The starting quarterbacks for both teams -- Cleveland's Brian Hoyer and Buffalo's E.J. Manuel -- were knocked out of the contest with knee injuries and Hoyer had an MRI scheduled for Friday amid concerns he had torn ligaments.
Brian Weeden, returning from a thumb injury, came in for Hoyer and completed 13-of-24 passes for 197 yards and a touchdown.
Buffalo's Jeff Tuel was not so fortunate while filling in for Manuel. Tuel went just 8-of-20 for 80 yards and it was his pass that Ward intercepted as Buffalo was trying to score a late touchdown that could have won the game.
In addition to the injuries suffered by the quarterbacks, Buffalo's leading receiver Steve Johnson left the game in the first half with a back injury and Cleveland running back Chris Ogbonnaya suffered a concussion.
Buffalo scored the game's first 10 points, fell behind at halftime 17-10, regained a 24-17 advantage in the third period and then saw the Browns score the final 20 points of the contest.
Cleveland improved to 3-2 and took a half game lead over both Baltimore and Cincinnati in the AFC North. Buffalo slipped to 2-3 and is in last place in the AFC East.
Despite scoring almost 40 points, the Browns managed only two offensive touchdowns -- those coming on a 1-yard run by Willis McGahee in the second period and a 37-yard throw from Wheeden to Josh Gordon in the third.
Cleveland came up with a key 79-yard punt return for a score by Travis Benjamin late in the half and Billy Cundiff kicked three field goals.
The highlight for the Bills came when C.J. Spiller ran 54 yards for a touchdown on Buffalo's opening possession of the second half to create a 17-17 tie.