Romo completed 22-of-27 pass attempts for 303 yards without an interception for the Cowboys (6-6), who logged their third win in four games.
The Cowboys erased a 27-24 deficit with 5:35 left in the game when Romo capped a seven-play, 86-yard drive with a 6-yard TD pass to Bryant, putting Dallas ahead to stay.
Morris Claiborne ensured the victory with a 50-yard fumble recovery for a touchdown that gave the Cowboys enough of a cushion to hold off the Eagles, who got a 98-yard punt-return touchdown by Damris Johnson with 31 seconds left.
Bryant finished with two TDs on six catches for 98 yards. Jason Witten also made six grabs for 108 yards for Dallas.
Bryce Brown followed up 178-yard rushing performance last week with 169 yards and two touchdowns on 24 carries Sunday for the reeling Eagles (3-9), who dropped their eighth straight.