Trailing 17-14 with just over three minutes left in the third quarter, Jacobs took a short pass from Eli Manning and rumbled 74 yards to put the Giants ahead 21-17.
Following a badly missed 42-yard field goal by Nick Folk and a Lawrence Tynes field goal, New York return man Domenik Hixon then scored on a 79-yard punt return to extend the lead to 14 and that was all she wrote.
"We gave up big plays we don't normally give up," Dallas head coach Wade Phillips said afterwards. "They didn't complete that many passes, but the ones they did were big plays for them."
The Giants defeated Dallas for the second time this season despite several enormous efforts from Jason Witten and Roy Williams.
Witten caught 14 passes for 156 yards and Roy Williams arguably had his best day as a pro, catching two touchdowns.
Miles Austin also had 10 receptions for 104 yards and another score.
Tony Romo finished the game completing 41-of-55 passes for 392 yards to establish single-game personal bests for attempts, completions and passing yards.
The Cowboys went up 10-0 in the first quarter and appeared to be poised to end the speculation that another December slump would doom their season.
But time after time the Giants responded.
After taking the 10-point advantage, it took New York just four plays to get back in it.
Going 74 yards in less than two minutes, rookie wide out Hakeem Nicks scored on a 21-yard pass from Manning and it was game on.
Moments later, New York was in position to take the lead as Mathias Kiwanuka forced a Barber fumble that was recovered by end Osi Umenyiora.
After a 24 yard return by the former Pro Bowler, Jacobs scored from one-yard out with 19 seconds left in the half, giving New York a 14-10 halftime lead.
The Cowboys return to action next weekend against San Diego at 4:15 PM in Cowboys Stadium.
NOTES: The Cowboys amassed 424 yards of total offense today to give them 4,737 through 12 games of the season – the most in team history through 12 games. ... Witten's 14 catches marked the second-highest single-game catch total in his career. His single-game career high was 15 receptions (at Detroit – 12/9/07).