Antecedents of the expression in question
Antecedents to the exact expression "sell [one's] soul to the devil" go back considerably earlier than the OED's circa 1570 citation. A search of Early English Books Online turns up several relevant instances.
From a 1509 translation of Antoine de La Sale, The Fyftene Joyes of Maryage:
I wyll that ye so god me saue and mende / Your hous frome hym forbede wolde and defende / With whome your frende me falsely hath accused / And you deceyued thus and sore abused / Now be it frely do I gyue or sell /
My soule vnto the foulest fende of hell / If euer he to me spake lesse or more
La Sale died in 1461, so if the English translation is at all close to the original, the idea goes back at least to the middle of the fifteenth century in France.
From Ihesus. The Floure of the Commaundementes of God with Many Examples and Auctorytees Extracte and Drawen as Well of Holy Scryptures as of Other Doctours and Good Auncient Faders (1510):
Another example of a man the whiche was so ylle instructe and correcte that he solde his soule and the deuyll bare it awaye with the body.
The disciple recyteth in his sermōs and sayeth that a man sayd vnto his compaygnons as they dranke at ye tauerne yf they byleued yt there was of sou∣les and that he byleued nothynge and yt he ne had none sene. And that the prestes and prechers had founde that there was of soules in helle for the gayne temporell And that he byleued nothynge. And hys compaygnons affermed yt after the fayth catholyke that there was of soules in hell and in heuen and that the soule of euery persone goeth vnto paradyse or in to hell or in to purgatorie after that it departeth from the body. After he demaūded yf ony wolde bye his soule. And that he sholde sel it. And one of his compaygnons sayd vnto him. I wyl bye it. And it was solde for a quarte of wyne the whiche was incontynent dronke. And there came the deuyl in the fourme of a man the whiche boughte the sayd soule of hym the whiche solde it and payed a quarte of wyne. And whan ye sayd wyne was dronke the deuyl said. Gyue me the soule yt I haue bought for it is myne. And those the whiche were present sayd. It is a thynge iuste that he the whiche byeth possede the thynge bought. And the accursed the whiche had solde his soule sayd that he had not solde the body. And the presence sayd. Whan yt ony selleth his hors he gyueth the coler or halter withall. And incontynent ye seller was estrangled of the deuyl. And before the presence rauisshed the sayd man & bare hym in to hel in body and soule.
From Here Begynne the the Lanterne of Lyght (1535[?]):
Howe shulde ye Iurryours scape the fyre of hell that (for a lytle money) wol dampne ye reckneuer whom and dysheryte trwe heyres of theyr iuste herytage for tho that wollen not say the truthe but yf they taken mede sellen Chryste that is truthe and bene worse then the Iewes for they slowen him whan he came to dye but now he reygneth vndedly wher shal thā your payne be? that woll say false wytnes for to catche auauntage of worldly wynnyge? ye sel your selfe your body & your soule into ye fēdes seruice yet may we se more encūbraūce of ye fēdes wirchyng for ther is none offycer temporal nor spūall but that he is redy whan he may to take gyftes of ye poore cōmyns and pyll them euer amonge and elles they shall no peace haue from greuousser oppressynge / as is taken of theyr beastes with corne and other vitayles and other payment get they none / but a whyte stycke tyll they haue lost halfe on halfe with moche more trauayle.
In this extract, "fēdes" is an archaic print spelling of fiends (or perhaps fiend's).
Pre-1570 instances of the expression itself
In addition finding texts that use other formulations to express the idea of selling one's soul to the devil, EEBO searches turn up two sources published before 1570 that explicitly use a close version of the modern expression.
From Thomas More, A Dialoge of Comfort Against Tribulacion (1534/1553):
Anthony. ... What goodes of this world can any mā imagine wherof the pleasure and commoditie could be such in a thousande yeare, as were able to recompence that intollerable payn, yt there is to be suffered in one yeare, or in one day, or one howre either: yea & thē what a madnes is it, for that poore pleasure of youre worldly goodes of so few yeares, to cast your self both body and soule into ye euerlasting fier of hel, wherof there is not minished the moūtenaunce of a moment by the lying there the space of an hundreth thousande yeares. And therfore our sauiour in fewe wordes concluded & confuted al these folyes of them, that for the short vse of this worldly substaunce forsake him and his fayth, and sell their soules vnto the deuill for euer, where he sayeth: Quid prodest homini sivniuersum mundum lucretur, anime vero• sue detrimentum patiatur? what auaileth it a mā if he wanne all ye whole world, & lost hys soule?
I am surprised that the OED didn't cite this instance of the expression, as More's book is well known and the wording he uses is very close to the form that English speakers use today.
And from Robert Horne, An Answeare Made by Rob. Bishoppe of VVynchester, to a Booke Entituled, The Declaration of Suche Scruples, and Staies of Conscience, Touchinge the Othe of the Supremacy, as M. Iohn Fekenham, by Wrytinge Did Deliuer vnto the L. Bishop of Winchester with His Resolutions Made Thereunto (1566):
Next, vnto him [Gregory], saith Nauclerus, was Syluester the second placed, by the Emperous appointment. Who being a Coniurer, had solde his soule to the Deuill for this promotion.
The notion of selling one's soul to an unspecified person or entity
It also seems worth noting that the notion of selling one's soul (though not specifically to the devil) appears in some fairly early English translations of the Bible and in related religious or religion-themed works. For example, from the Book of Ecclesiasticus in Biblia the Byble (1535):
There is not a more wicked thinge, then to loue moneye. And why? soch one hath his soule to sell: yet is he but fylthie dōge whyle he lyueth.
And in "a table of the principle matters conteyned in the Bible" (essentially an index), in a version of the Bible printed for Thomas Barthlet in 1539, the editor includes this entry:
Couetousnes. ... | There is nothynge more wycked then the couetous man: for he selleth his owne soule Eccle .x. b.
From Antoni Gilby, "An Admonition to England and Scotland to call Themselves to Repentance," in John Knox, The Appellation of Iohn Knoxe from the Cruell and Most Iniust Sentence Pronounced Against Him by the False Bishoppes and Clergie of Scotland (1558):
For the communes did continew in malice, and rebellion, in craft and subtiltie, notwithstanding all lawes, that could be deuised for reformation of abuses. The merchants had their own soules to sell for gaines, the gentlemen were becomme Nērods and Gyants, and the nobilitie and coūsile would suffer no rebukes of Gods messēgers thogh theyr offenses were neuer so manifest.
And from A Newe Mery and Wittie Comedie or Enterlude, Newely Imprinted, Treating vpon the Historie of Iacob and Esau (1568):
Esau. But the best pottage it was yet that euer was. / It were sinne not to sell ones soule for such geare.
Discussions (in published English writings) of selling one's soul to the devil antedate the earliest instance that the OED cites of the notion of selling one's soul to anyone or anything. That instance, noted in linguisticturn's answer, is from 1570 and refers broadly to "sell[ing] their soules & all for monye"—without specifying the buyer. The earliest instance that the OED cites (according to linguisticturn) of "sell [one's] soul to the devil" was from more than a century later, in 1677.
Searches of the Early English Books Online database, however, find books from as early as 1509 and 1510 discussing the notion of selling one's soul to diabolical forces (fiends or devils). Moreover Thomas More, writing in 1534, uses essentially the modern phrasing of the expression except that he replaces "to" with "unto": "sell their souls unto the devil." It seems to me that More's use of the phrase deserves a place in the OED's account of its earliest occurrences.