The Bucks had lost five consecutive games.
Tony Parker scored 22 points for San Antonio, which has lost all four of its road games while winning all six of its home contests.
Richard Jefferson's tying 3-point attempt hit the back of the rim at the buzzer.
Jackson was obtained from Charlotte on June 23 in a deal that also included Sacramento. He struggled with his shot the first eight games, making only 32 percent of his field-goal attempts.
He had no such problems against the Spurs, making 12 of 17 shots, including a 3-for-4 night from 3-point range. He also hit all seven of his free-throw attempts while playing 42 minutes.
Trailing 104-103, Parker set up San Antonio's offense with just under a minute left looking for the go-ahead basket. Jennings reached around his back and knocked the ball away to Jackson.
Jackson made a quick pass to a streaking Jennings for the easy fast-break dunk, giving the Bucks a 106-103 lead with 50.1 seconds left.
Duncan then was called for an offensive foul on Bogut with 35 seconds left.
After a jump ball went to the Spurs with 15 seconds remaining, Parker worked the 3-point line, but was unable to shake Jennings and passed to Jefferson. The forward's shot went off the back of the rim and was rebounded by Bucks rookie Jon Leuer as time expired.
The Bucks took the lead for good, 93-92, on Jackson's 14-foot jumper with 6:53 to go.
Jefferson's 3-pointer in the closing seconds of the first half pulled the Spurs into a 59-59 tie at the break.
NOTES: Spurs G T.J. Ford strained his left hamstring in the first quarter and did not return. . Bucks G Mike Dunleavy will miss another two to four weeks due to a groin injury. He has only played in three games. . Bucks F Luc Richard Mbah a Moute (right knee tendinitis) and G Beno Udrih (left shoulder sprain) both missed the game. . It was the only time the teams play this season.