There are a number of other noun-verb pairs that follow this sound pattern (sometimes with a difference in spelling): such as house, house; advise, advice; devise, device.
This is mentioned in the answers to the following question: Why don't "-use" verb-noun pairs obey initial stress derivation?
The origins of voicing fricatives in verbs
In general, this type of alternation originates from sound changes earlier in English that turned voiceless fricative sounds (such as [s, f, θ]) into voiced fricative sounds (such as [z, v, ð]) in some environments where they were between voiced sounds (generally between two vowel sounds or between a voiced consonant such as /l/ or /r/ and a vowel sound). Examples of native English words affected by this process: the verbs house, halve, sheathe which are related to the nouns house, half, sheath.
However, this question deals with a word that comes from French, and it turns out that French also underwent a sound change turning [s, f] into [z, v] in similar conditions.
At the time these sound changes applied, verbs in both of these languages tended to be marked with suffixes that started with vowels. So in a number of cases you have a noun that ends in a voiceless fricative, and a related verb which has a voiced fricative due to the presence of the vowel-initial verbal suffix.
Due to later sound changes, English verbs don't have vowel-initial verb suffixes in most of their forms today (usually only the gerund-participle suffix -ing has a vowel sound), but the voicedness that these vowels caused remains (and English kept voiced final fricatives in the stems of verbs taken from French, even though the French verb endings were generally discarded).
The etymology of the noun "use"
According to the Oxford English Dictionary, the noun "use" comes from an Anglo-Norman/Old French word that could be either feminine (with an -e suffix) or masculine (ending with the consonant s); it appears that the masculine form (which would have been pronounced with voiceless [s]) contributed more to the English pronunciation. The modern pronunciation with /s/ may also have been reinforced by the pre-existing pattern of voiced and voiceless consonants in English. At least some nouns, such as "wreath", end in voiceless fricatives today even though etymologically they had voiced fricatives ("wreath" comes from Old English "wriþa"). An example of this happening in a noun from French (although one that was borrowed much more recently than "use") is the noun "dose," which has a feminine suffix -e and a /z/-sound in French, but which has come to be pronounced with /s/ in English.
The etymology of the verb "use"
On the other hand, the OED says the verb "use" comes from an Anglo-Norman/Old French verb where the "s" came before a verbal suffix "-er(e)". It would therefore have been pronounced as voiced [z]. "Using" is an inflected form of the verb, which explains why it has the same consonant sound. (I am not aware of any verbs in Modern English that modify consonant sounds between the bare form/infinitive and the -ing form/gerund-participle). The noun "user" is derived from the verb, so it is also expected to have the same consonant sound as the verb.
The strange history of the noun "usage"
The noun "usage" is interesting. It actually is pronounced with both /z/ and /s/. The /z/-pronunciation is easier to explain from an etymological perspective, as the "s" is between vowels. The OED suggests that the /s/-pronunciation is due to influence from the pronunciation of the noun "use", which is the same part of speech and has a similar meaning.
This is just speculation, but I'd guess it was easier for speakers to change from /z/ to /s/ in "usage" than in "use" (v) because there are many commonly used verbs with voiced final fricatives that correspond to nouns with voiceless final fricatives, but there are few common examples of voicing before the suffix -age. Nouns ending in -age may be apparently related to nouns (leafage), or verbs (cleavage), or neither (visage), or they may be ambiguous about whether they're related to a noun or a verb (dosage).
Thus, it's easy to analyze "usage" as being composed of "use" (n) + "-age," and since modern speakers don't voice fricatives before the suffix "-age" by default, this analysis suggests using a voiceless fricative /s/.