DeMarco Murray had two touchdown runs to help the Cowboys move ahead of the Eagles into first place in the NFC East.
"Just came out poised. I was focused," Bryant said. "We knew that this game was going to be big."
The Cowboys took a 21-0 lead early in the second quarter before Philadelphia rallied behind Mark Sanchez.
Chris Polk had touchdown runs of one and five yards, and Darren Sproles ran in from the one to give the Eagles a 24-21 lead late in third quarter. But Romo and the Cowboys answered with two touchdowns in a span of 2 minutes 51 seconds.
Murray scored on a two-yard run to cap a 78-yard drive. After J.J. Wilcox intercepted a Sanchez pass, the Cowboys started the fourth quarter at the Eagles 42. Romo hit Bryant in stride for a 25-yard touchdown pass.
Bryant finished with six receptions for 114 yards.
"Having a guy like Dez makes it easy to throw to spots that he can go get it," Romo said. "Just lucky to have him."
Romo threw for 265 yards and Murray had 81 yards in 31 carries.
Sanchez had 252 yards passing and two interceptions. LeSean McCoy was held to 64 yards rushing.
The Eagles had won 10 consecutive regular-season games at home before losing to Seattle and Dallas for their first consecutive losses since Weeks 7 and 8 in 2013.
"We can't turn the football over and expect to win games," Coach Chip Kelly said. "We're not coaching it and we're not doing the right things."