Devin Harris scored 27 and Vince Carter had 26 for the Nets, who have lost five straight. Carter threw up a 3-pointer with 0.2 seconds left, but the shot was waived off before the ball left his hand.
Carter and Nets coach Lawrence Frank rushed to halfcourt to protest with officials, but the game was over. Carter pounded his hands together in frustration, untucked his shirt and walked off the court.
Spurs coach Gregg Popovich had already walked his players off the court before Carter's shot, but had to return after officials reviewed the film and found there was a sliver of time left - but not enough for Carter to catch-and-release.
Tony Parker scored 17 points and Roger Mason had 14 for San Antonio. Finley finished with 11 after drilling his open 3 from the corner, which sent the San Antonio crowd screaming and jumping to their feet after a tight fourth quarter.
Duncan shot 13-of-21 from the floor. He also had five assists and four blocks.
The Spurs won't have much time to celebrate another close victory.
They're heading West to begin their toughest road trip of the season, starting Sunday with the Los Angeles Lakers in a rematch of their 112-111 thriller this month that San Antonio snatched in the final seconds.
Then it's on to Utah and Phoenix before coming back for one last home game - New Orleans, nipping at San Antonio's heels in the Southwest Division - before clearing out of town most of February for the city rodeo.
The Nets arrived losers of four straight - the Celtics clobbered them twice - but kept their Spurs on their toes from the start.
Carter set the tone with two quick jumpers and a hard, one-handed dunk after wrapping around Duncan that ushered the first "oohs" of the game. Carter, who finished third in All-Star voting among guards in the East, finished the first half with 15 points.
The Nets opened the second quarter on a 13-0 run with Duncan and Tony Parker on the bench for the Spurs, and Harris hit a pull-up 3-pointer on a fast break with 28.8 left to give New Jersey a 50-44 halftime lead.
The Nets lost their fifth straight, but the schedule gets easier. New Jersey finishes the latter half of its four-game road swing at cellar-dwelling Memphis and Oklahoma City.
Carter shot 9-of-18 from the field and rookie Brook Lopez had 15 for New Jersey.
Notes: During a timeout midway through the second quarter, players from both teams paused to applaud thousands of military personnel in attendance. ...For a small-market team, the Spurs apparently have a lot of fans - or their fans know how to stuff the All-Star ballot box. Spurs F Bruce Bowen, playing fewer minutes this season than any point in his 13-year career, finished third among Western Conference forwards with 1.39 million votes.