Rice finished with 63 yards on 16 carries while Flacco completed 17-of-26 passes for 234 yards as the Ravens (5-1) notched their fourth straight victory.
Jacoby Jones returned a kickoff an NFL record-tying 108 yards in the third quarter for Baltimore, which only secured the win after turning back a late Cowboys rally.
Tony Romo (25-of-36 passing, 261 yards) hooked up with Dez Bryant twice for touchdowns, but Bryant couldn't make the catch on a potential game-tying 2-point conversion with 32 seconds left following his second TD grab.
He finished with 13 receptions for 95 yards while DeMarco Murray rushed for 93 yards on 14 carries to lead Dallas (2-3).
Notable deaths of 2014 [PHOTOS]