After Tayshaun Prince tied it with a put-back with 9.8 seconds left, Hinrich caught a pass at the top of the key and got fouled by Carlos Delfino on a 3-point attempt.
Hinrich missed the first free throw but hit the next two.
After a timeout, Prince caught the inbounds pass and launched a short pull-up jumper over Luol Deng as time expired.
It was the fourth straight win for the Bulls and their 11th in 14 games. Chicago trails Eastern Conference leader Detroit by three games and second-place Cleveland by a half game, with the Cavaliers visiting the United Center on Saturday.
Ben Gordon led the Bulls with 25 points, and Ben Wallace grabbed 19 rebounds against his former team.
Prince scored 26 points for Detroit.
With a 77-65 lead and about five minutes remaining, the Bulls appeared to be in control. But the Pistons were within 81-79 with 48 seconds left after Chauncey Billups drove by Hinrich.
Rasheed Wallace then stole the ball from rookie Tyrus Thomas with 35 seconds left, and Prince put back Delfino's missed jumper to tie it at 81.
Thomas had arguably his best game, finishing with 13 points and 10 rebounds, and Hinrich scored 15. Luol Deng finished with 16 points after scoring a career-high 38 in Monday's win over Portland.
The Pistons were without stars Richard Hamilton and Chris Webber, who were back home with the flu.
Rasheed Wallace scored 19 points, but picked up his 19th technical after Ronald Murray was called for a foul with 6:45 left in the third quarter. The next one will result in another mandatory one-game suspension for Wallace, who served one Sunday when Detroit visited Milwaukee.
With the top two teams in the conference in town this week, the Bulls had a chance to see where they stack up and - more importantly - boost their playoff seeding. Although the odds of catching the Pistons remain slim, the Cavaliers certainly are within reach.
The Bulls appeared to be in good shape early in the fourth. They started it with a 10-2 run, turning a 61-60 deficit into a 70-63 lead, and Gordon and Ben Wallace played big roles.