Giants running back David Wilson, who also fumbled on New York's first running play in the game, was stripped early in the second half by Nick Hayden. Church scooped up the ball and ran into the end zone.
Without a fourth-quarter comeback, Eli Manning and the Giants would no longer be undefeated in Cowboys owner Jerry Jones' $1.2 billion showplace stadium.
They were 4-0 in the building, including a victory in the first regular-season game there in 2009.
Manning had two TD passes to Victor Cruz, including an 18-yarder with 1:44 left in the third quarter.
The Giants already had five turnovers (two fumbles by Wilson, two interceptions by Manning and a botched punt return).
They hadn't had five turnovers since an October 2011 game against Seattle.
Dallas' defence forced only 16 turnovers all last season (seven interceptions, nine fumbles).
The Cowboys had three in their first 6 1/2 minutes playing in new defensive co-ordinator Monte Kiffin's 4-3 scheme.
A play after a possible pass interference call on Dez Bryant didn't draw a penalty and the Cowboys punted, the ball hit Giants blocker Trumaine McBride and was recovered by rookie DeVonte Holloman at the Giants 16.
That set up Romo's 4-yard TD throw to Witten, the eight-time Pro Bowler who had only three scores last season, for a 27-10 lead.
Romo threw a 15-yard score 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.
Manning hit 12 of 20 passes for 249 yards in the first three quarters, and Cruz had eight catches for 108 yards.
Romo was 30 of 42 for 241 yards. Miles Austin had nine catches for 59 yards and Witten had seven catches for 66 yards.
Witten's first TD catch with 3 minutes left in the first half made it 13-3 and came with linebacker Mark Herzlich trying to cover him in the end zone. The grab put him over 9,000 yards receiving. He also had a 21-yard catch on the previous play.
But Manning, who threw two interceptions in the first 6 1/2 minutes of the game, found a wide-open Cruz for a 70-yard touchdown after snapping the ball right before the two-minute warning of the first half.
The Giants had been forced to call a timeout before taking the first snap on that drive, and then had a false start penalty out of the timeout.
On the first play of the game, Manning was trying to complete a short pass to Wilson. DeMarcus Ware, the perennial Pro Bowl linebacker now playing defensive end, reached out and grabbed the throw, setting up the Cowboys at the Giants 15.
After Manning's second interception, this one by new Cowboys safety Will Allen, Dallas had third-and-11 at the Giants 14.
Romo then threw a pass toward rookie Terrance Williams that instead deflected off Prince Amukamara into the arms of Ryan Mundy, who raced 91 yards. He was caught from behind by running back DeMarco Murray at the 1.
Amukamara and Mundy had a less pleasant connection with Williams late in the first half. The defensive backs collided helmet-to-helmet while tackling Williams after a completed pass. Amukamara sustained a concussion and didn't return, while Mundy was cleared to return after halftime.
After Mundy was run down at the 1 on his interception return, Wilson was then stuffed for no gain before Manning was sacked, and the Giants had to settle for a 20-yard field goal by Josh Brown.
AP NFL website: www.pro32.ap.org