CLEVELAND — When C.J. Spiller got hurt, the Buffalo Bills lost the NFL’s leading rusher, most of their momentum, and another major offensive threat.

Ryan Fitzpatrick wouldn’t let them lose the game.

Fitzpatrick threw three touchdown passes and the better-than-advertised Bills overcame the loss of Spiller to a shoulder injury and beat the Cleveland Browns, 24-14, Sunday to snap an eight-game road losing streak.

Fitzpatrick finished 22 of 35 for 208 yards as the Bills (2-1) turned to their passing game after Spiller, who has been filling in for the injured Fred Jackson, went down in the first quarter with an injury that will likely sideline him a few weeks.

The Bills won outside Buffalo for the first time since their 2011 opener at Kansas City.

‘‘We got the monkey off our back,’’ Fitzpatrick said. ‘‘We always felt confident. This is a new year, a new team, but it was a good team win on the road. These are the most satisfying wins.’’

Spiller, who came into Sunday leading the league with 292 rushing yards, scored on a 32-yard screen pass from Fitzpatrick, who also threw a 9-yard TD pass to T.J. Graham in the first quarter and hooked up with Stevie Johnson from 9 yards in the fourth.

Bills coach Chan Gailey does not expect Spiller to be back for next week’s home game against the Patriots.

‘‘I don’t believe it’s long, long term,’’ Gailey said. ‘‘If he missed a week that’s long for me.’’