Jon Scheyer scored 15 points and Kyle Singler added 14 and grabbed 12 rebounds for the Blue Devils (8-0), who needed big plays from Paulus down the stretch to extend their winning streak over Davidson to 21 straight.
Davidson (3-3), which took top-ranked North Carolina to the wire just over two weeks ago, rallied from down 16 in the first half and 12 with 5 minutes to play, getting within 70-65 with 2:37 left.
Paulus then hit a 3-pointer, and after Stephen Curry hit a three at the other end, Paulus' baseline fadeaway with 32 seconds left gave Duke a 75-68 lead.
Paulus then stole the ball from Jason Richards, and Duke held on with Henderson's two free throws with 5.3 seconds left putting it away.
Curry scored 20 points despite playing the final 12 minutes with four fouls, and Richards added 14 points and 11 assists for Davidson.
Poor shooting did in Davidson again in its 23rd straight loss to a ranked team. The Wildcats hit eight of 24 threes, after shooting 4-of-22 from long range against the Tar Heels. Davidson also missed seven of 12 free throws.
Duke had won its first seven games by an average of 15 points, including an 82-58 rout of No. 20 Wisconsin on Tuesday. But despite being far superior in athleticism, the Blue Devils never got comfortable against Davidson, which won a school-record 29 games last season before losing to Maryland in the first round of the NCAA tournament.
Duke, though, was able to hit 11 3-pointers to offset 16 turnovers and win its 42nd straight game against Southern Conference teams, in part thanks to foul trouble from Davidson's star.
Curry, the son of former NBA three-point specialist Dell Curry, picked up his second offensive foul with 8:39 left in the first half. With him on the bench, Duke immediately scored 10 straight points in a 17-4 run.
The Blue Devils eventually took a 40-24 lead on Scheyer's 3-pointer and three free throws.
The Wildcats got back in the game by going on an 8-0 run to start the second half, with Curry's straightaway 3 cutting Duke's lead to 43-40.
But Curry would pick up his fourth foul - on another charge - with 11:58 left, and Henderson's alley-oop dunk and driving layup got Duke's cushion back to 10.
Duke improved to 85-17 against Davidson.