Stephen Jackson had 12 of his 24 points in the fourth quarter, including a key 3-pointer with 46 seconds left.
Ben Gordon led Chicago with 29 points and Andres Nocioni added 28.
Matt Barnes scored the go-ahead basket for Golden State, grabbing a loose ball as Nocioni tried to save it, putting the Warriors up 106-104 with just over two minutes to go.
Joe Smith's basket cut it to one before Jackson dropped in two free throws with 1:20 left, making it 109-106.
After Gordon's two free throws again made it a one-point game, Jackson took a pass from Davis and made a 3-pointer from the deep corner, putting the Warriors up four.
After a miss by Nocioni, Davis made two free throws. Then Jackson hit two more and the Warriors had a hard-earned road win against the disappointing Bulls.
Golden State outscored the Bulls 35-27 in the fourth quarter.
Davis was 5-for-5 on 3-pointers in the third quarter, scoring 19 points in the period as the Warriors erased a 12-point deficit.
With Chicago ahead 69-57, Davis went to work, making a nice feed to Monta Ellis for a basket and then hitting two 3-pointers. Jackson hit another one and suddenly the Warriors were within three.