(Reuters) - A banged-up Dallas offense mustered a late game-winning drive to help the Cowboys grind out an 18-16 win over NFC East rivals Washington on Monday.

Dallas quarterback Tony Romo, who sustained fractured ribs and a lung injury last week, led the Cowboys 64 yards down the field in the closing minutes to set up the winning field goal.

Washington had one last possession but Dallas linebacker Anthony Spencer stripped quarterback Rex Grossman of the ball in the final seconds to give the Cowboys (2-1) the win. It was the Redskins (2-1) first loss of the season.

"This is a great win for us. It was a hard, hard game," Romo told reporters. "We had a million mistakes tonight. We're lucky to get out of here with a win. We're going to get it fixed -- we can't have that happen."

Romo, who had to deal with his center snapping the ball early on several occasions, threw for 255 yards with one interception while running back Felix Jones gained 115 yards on the ground despite playing with an injured shoulder.

Dallas were also without top receiver Miles Austin while Jason Witten and Dez Bryant also played hurt.

"Injuries are a part of it, they have been for a long time. You can never use that as an excuse," said Cowboys coach Jason Garrett. "We had guys picking us up. I thought the defense was outstanding."

The first half turned into a kicking contest with the Redskins' Graham Gano and Cowboys' Dan Bailey slotting over three apiece for a 9-9 tie at half-time.

The Redskins took a 16-9 lead in the third quarter on a one-yard touchdown pass from Grossman to Tim Hightower but Bailey brought Dallas back, kicking three more field goals including the game-winner from 40 yards with 1:47 remaining.

The winning kick was set up after Dallas converted on third and 21, Romo finding Bryant on a 30-yard pass.

Washington's offense sputtered in the fourth quarter and they were forced to punt three times. (Writing by Jahmal Corner in Los Angeles; Editing by Peter Rutherford)