The Texans (4-7) had lost five straight road games this season and hadn't won away from home since Nov. 4, 2007, at Oakland. But all it took was a weekend in Cleveland to fix their travel woes.
Rosenfels, a career backup, had his best game since replacing the injured Matt Schaub three weeks ago. He went 24 of 32 and hooked up with Kevin Walter on a 17-yard TD pass in the first quarter. However, Rosenfels nearly let the Browns back in it with a pair of interceptions in the second half.
Andre Johnson had 10 catches for 116 yards as the Texans also ended a three-game losing streak.
A disastrous season got even worse for the Browns (4-7), who lost their third straight at home and watched their ever-faithful fans leave by the thousands in the final minutes. Unlike the previous two losses, the Browns didn't blow a double-digit lead as they fell behind 10-0 early and never recovered.
Quarterback Brady Quinn was replaced in the second half of his third start after throwing two interceptions, the second a killer inside Houston's 10-yard line.
Quinn, who was given the job following Derek Anderson's early-season struggles, went 8 of 18 for 94 yards. Quinn was playing with a broken right index finger on his throwing hand, but the Browns never announced any injury when coach Romeo Crennel pulled the former Notre Dame star with Cleveland down 16-6 in the third quarter.
Anderson couldn't rally the Browns and was again victimized by wide receiver Braylon Edwards, who dropped one that was right in his hands and couldn't pull down a leaping attempt in the end zone that could have cut the margin to a field goal in the fourth quarter.
Anderson went 5 of 14 for 51 yards, threw an interception and lost a fumble.
Browns running back Jamal Lewis had two uncharacteristic fumbles and kicker Phil Dawson, who booted a game-winning 56-yarder in Buffalo on Monday, missed from 39 in the fourth when the Browns still had a chance.
The loss came a few days after general manager Phil Savage had to apologize for a profane e-mail he sent to a fan following the win over the Bills.
Rosenfels connected with Walter on a perfectly thrown 17-yard TD pass to give the Texans a 7-0 lead.
Houston took the opening kickoff and methodically marched 79 yards in 14 plays, eating up more than eight minutes on the clock before Rosenfels threw to the left corner where the 6-foot-3 Walter made a leaping, twisting catch.
The Texans got the ball right back when Jacques Reeves knocked the ball loose from Lewis, whose fumble was recovered by DeMeco Ryans at Cleveland's 40.
Houston, though, couldn't turn the field position into a touchdown and settled for Brown's 31-yard field goal to go up 10-0.
Cleveland got two field goals from Dawson to close within 10-6 before Brown kicked another 31-yarder with three seconds left to make it 13-6 at halftime.
Brown kicked a 38-yarder for the only points in a sloppy second half by both clubs.
