You're looking for two different idioms. I don't know one idiom that covers both situations in English.
- Using the personal connection:
Know the right person -- note, this might not be someone high up.
Example: He can get that ticket fixed for you. He knows the right person/people.
- Paying someone off with a bribe:
(a) Grease someone's palm -- Give someone money in exchange for a favor; also, bribe someone. American Heritage Dictionary of Idioms, published at the Free Dictionary.
(b) Pay someone under the table -- this is an alternative but it needs some context for it to be clear that there is a bribe. You could use this one occasionally for variety.