The Seahawks (5-1) gave up touchdowns on Cleveland's first three possessions before storming back.
Wilson connected with Jaron Brown for two scores, and Seattle's cool quarterback scampered 16 yards for a TD. The Seahawks last won their first three road games in 1980, when they went 4-12.
The Browns (2-4) were hurt by their own miscues. They couldn't overcome four turnovers, a blocked punt or quarterback Baker Mayfield's hip injury and fell to 0-3 at FirstEnergy Stadium.
Mayfield threw three interceptions, raising his NFL-leading total to 11. Trying to bring the Browns back late, Mayfield, who went to the locker room in the third quarter to be evaluated, was picked off by Seattle's K.J. Weight with 2:41 remaining.
The Seahawks defenders celebrated the game-clinching turnover in front of Cleveland's Dawg Pound section and were pelted with cans and other debris.
Wilson then ran out the clock as the Seahawks moved to 5-1 for the third time.
Carson's short TD burst capped an efficient 79-yard drive by Wilson, who showed his usual poise and precision after the Browns had taken a 28-55 lead on Nick Chubb's 3-yard TD run.
Wilson finished 23 of 33 for 299 yards and stayed error-free. He has yet to throw an interception this season, and to this point, the eight-year veteran is outplaying every other QB in the league.
Mayfield, on the other hand, is struggling. He completed 22 of 37 passes for 249 yards, but he forced several throws and made a deliberate attempt to get the ball to Odell Beckham, who finished with six catches for 101 yards.
Carson gained 124 yards on 24 carries for the Seahawks, who lost tight end Will Dissly in the first half with an Achilles injury.
Chubb had 122 yards on 20 attempts.
Even after the Browns scored on their first three drives and opened a 20-6 lead, Wilson wasn't rattled.
He hung in the pocket and took a big hit from linebacker Joe Schobert while completing a 17-yard TD pass with 22 seconds left in the half to pull the Seahawks within 20-18. Seattle's 2-point conversion failed, but the Seahawks were back in it after the Browns got greedy moments earlier.
Instead of running the ball, first-year coach Freddie Kitchens called a pass play and Mayfield forced one into the end zone that was behind Jarvis Landry, tipped and intercepted by safety Tedric Thompson.
Mayfield's first career TD run put the Browns up 14-6 in the first quarter. With no receivers open, Mayfield, who hasn't been running as much as he did as a rookie, took off and scored without the Seahawks laying a hand on him.
The TD scamper came just two plays after Mayfield converted on fourth-and-7 with a 27-yard pass to Landry.
Mayfield didn't participate in the pregame coin toss, his absence notable since it came just days after San Francisco cornerback Richard Sherman said the quarterback disrespected him by not shaking hands prior to kickoff.
Sherman later backtracked after video showed Mayfield shaking hands with him.
Seahawks: Dissly went down untouched after Wilson overthrew him on a pass in the second quarter. Dissly's 2018 season ended after four games when he tore his right patellar tendon.
Browns: Mayfield was escorted to the locker room, but came back. ... DT Daniel Ekuale sustained a head injury in the closing minutes.
Seahawks: Host Baltimore on Oct. 20.
Browns: Have a bye next week before traveling to New England on Oct. 27.
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