Schwartz has lost nine straight games in December or January, leading to his job being in jeopardy.
Detroit needed some help to stay in postseason contention and got it when Pittsburgh won at Green Bay, but the Packers' loss turned out to be moot when the lowly Lions fell.
The Giants (6-9) overcame Eli Manning's interception late in regulation and Andre Brown's fumble on the opening possession in overtime to win for the second time in five games.
With nothing to lose, New York went on fourth-and-7 from the Lions 42. Manning connected with Jerrel Jernigan to set up the winning kick — the second straight by a visitor at Ford Field.
Manning was 23 of 42 for 256 yards with a TD pass to Jernigan — his first career score — and an NFL-high 26th interception.
Stafford was worse: 25 of 42 for 222 yards and two interceptions, the second returned by Will Hill for a TD that to tied it at 20 with 4:57 left in the lackluster game.
Stafford's first interception and Reggie Bush's career-high fourth fumble in the first half let the offensively challenged Giants take a 13-3 lead at halftime.
New York started the game with an NFL-high 39 turnovers and Detroit ranked third in the dubious category with 31.
The Giants didn't have a turnover in the first half, while the Lions gave away the ball twice in the first half. It was costly each time.
Manning's 20-yard pass to Jernigan gave the Giants a 10-3 lead after Bush's fumble. Jernigan, a third-year receiver, took advantage of more playing time with Victor Cruz out after left knee surgery Thursday. Detroit cornerback Bill Bentley was beaten by Jernigan on the play and hit by safety Louis Delmas, who gave his teammate a concussion.
Brown's 52-yard field goal put New York ahead by double digits with 9 seconds left in the half. The kick came two snaps after Lions safety Don Carey got a delay-of-game penalty.
Detroit's Jeremy Ross returned the first punt of the second half 50 yards to set up a 1-yard TD run by Joique Bell to make it 13-10.
Lions defensive tackle Nick Fairley sacked Manning in the end zone to pull the Lions within a point late in the third quarter.
Detroit went ahead for the first time on the ensuing possession. Rookie running back Theo Riddick ran for a 2-yard TD and Fauria caught a pass for a 2-point conversion to give the Lions a 20-13 lead.
The Lions, though, blew it — and their chance to possibly win their first division title since 1993.