Cruz, who missed most of 2014 with a knee injury and all of last season with a bad calf, broke free in the end zone for a 3-yard TD with 6 minutes remaining, prompting his trademark salsa dance.
"You couldn't even describe what was going through my body," said Cruz , who had four catches for 34 yards. "Wanted obviously to do the dance. I'm sure I've got that down pat.
"Everybody just pulled for me each and every day, each and every week as we got closer to Week 1. And to score a touchdown. You just couldn't make up a story any better than that."
The game ended with Dallas in position for a roughly 57-yard field goal, but Terrance Williams didn't get out of bounds after a catch and the clock ran out before Prescott could spike the ball.
"The guy was trying to make a play," Prescott said of Williams. "You never want to knock a guy trying to make a play."
New York beat Dallas in an opener for the first time in nine tries.
Elliott was held to a 2.5-yard average (51 yards on 21 carries), although he had his first touchdown, and the only one for Dallas. Prescott was 25 of 45 for 227 yards.
Copyright 2016 by The Associated Press