Tony Romo shook off hits to the ribs that knocked him out for the final Dallas play of the first half and threw a pair of touchdown passes to Jason Witten.
Manning threw three touchdown passes to Victor Cruz and four overall, and he had New York in position for a comeback victory despite the flurry of mistakes.
But with the Giants trailing 30-24, Manning's pass went off the hand of running back Da'Rel Scott into Carr's arms.
The cornerback beat Manning down the sideline with 1:50 left for Dallas' second defensive touchdown. Barry Church returned a fumble 27 yards for a touchdown in the third quarter.
"It's infectious. It's contagious," Cowboys coach Jason Garrett said. "We challenged them at halftime. We had three, let's get three more."
The Cowboys had only 16 turnovers all last year, switched to a four-man front with new defensive coordinator Monte Kiffin and swore all preseason that taking the ball away was the foremost thing on their minds. They proved it from the first snap of the season.
DeMarcus Ware intercepted Manning's first pass, and the Giants turned it over on their first three possessions. The Cowboys didn't take advantage of the early gifts, but built a big enough lead to hold off a New York comeback once the Giants finally stopped turning it over.
It matched the most turnovers for the Giants since 1987, when they had seven against New Orleans. And it was the most Dallas has forced since 2003.
"Just tells you our system is so much built on attitude," said defensive line coach Rod Marinelli, who was defensive coordinator for Chicago last year when the Bears led the NFL with 44 turnovers. "We had a lot of guys play tonight who jumped in and helped us. It's just team defense."
Church scored on the fourth New York turnover -- the second fumble from new starting running back David Wilson, who lost control of the ball on a strip by Nick Hayden just before hitting the ground. Church scooped up the ball and ran it in.
The Cowboys went up 27-10 in the third quarter after the fifth miscue -- a punt that bounced and hit Trumaine McBride while he was trying to block for the return. Rookie DeVonte Holloman won a scramble for the loose ball, and Romo threw a 4-yard scoring pass to Witten.
"I just think it's pretty exciting to see what our team defensively has been able to do here, in training camp, preseason and into the season," Romo said.
When the mistakes stopped, the Giants made it a game again. New York went 80 yards, capped by Cruz's second score on an 18-yard pass, and Cruz scored again from the 10 to pull the Giants to 30-24 midway through the fourth quarter.
New York got the ball back, but the Cowboys forced a quick punt. After Dallas again failed to move the ball, Carr came through with the clinching interception.
"You're not going to win the game turning the ball over five times and muffing a punt," Giants coach Tom Coughlin said. "I'm totally, totally disappointed, embarrassed about that football. That's sloppy, sloppy football."
The Giants scored with 11 seconds left on Manning's fourth touchdown, a 4-yarder to Brandon Myers. But Dallas recovered an onside kick.
Manning was 27 of 42 for 450 yards and had three 100-yard receivers in Cruz (five catches, 118 yards), Hakeem Nicks (five for 114) and Rueben Randle (five for 101)
Romo was 36 of 49 for 263 yards, and Miles Austin had 10 catches for 72 yards. Witten, who had only three touchdowns last year, had eight grabs for 70 and became the third tight end in NFL history with 9,000 yards receiving. DeMarco Murray had 86 yards rushing for Dallas.
Romo threw a 15-yard TD pass to Witten before halftime, and Dan Bailey also had two field goals (30 and 38 yards) by then for a 13-10 lead.
In the final minute of the first half, Romo was hit in the ribs and had the wind knocked out after being crushed between Mathias Kiwanuka and Justin Trattou after throwing an incompletion. Kyle Orton came on to finish the half, but Romo was back after halftime.
Cowboys owner Jerry Jones said Romo didn't receive any treatment, and that the medical staff cleared him to play.
"One tough player, one tough man," Jones said. "He will go out and play in pain and play hurt."
After Manning's second interception, this one by new Cowboys safety Will Allen, Dallas had third-and-11 at the Giants 14.
Romo threw a pass toward rookie Terrance Williams that instead deflected off the Giants' Prince Amukamara into the arms of New York teammate Ryan Mundy, who raced 91 yards. He was caught from behind by Murray at the 1.
Game notes
The crowd of 85,348 was significantly smaller than the NFL regular-season record crowd of 105,121 that watched the first Giants game at the stadium in its 2009 debut. ... Amukamara was knocked out of the game with a concussion after a helmet-on-helmet collision with Mundy, who was cleared to return.
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