He hit 11 of 20 passes for 212 yards, didn't turn the ball over and converted two fourth-down plays with his feet.
"I'm aware that it was the first game, and in that respect, it's not so bad," said Mayfield, the Heisman Trophy winner from Oklahoma.
Mayfield said he didn't experience any jitters during his preseason debut, but did acknowledge that he was glad to have the first game in the books.
"The whole routine was different for me," Mayfield said. "It was the first time going on the road with a new team, first time in the NFL. Just some new things for me. I wouldn't say it was anything nerve-racking, but a lot of firsts today that I'm glad I got under my belt so I can get more comfortable as I go down the road."
Mayfield found tight end David Njoku on a 10-yard TD pass to cap a 14-play, 72-yard drive on his second series. Mayfield finished his debut with a completion on a 54-yard, slant-and-run score to fellow rookie Antonio Callaway.
Cleveland coach Hue Jackson said Mayfield's debut success didn't surprise him.
"I thought he did some good things out there," Jackson said. "I'm honestly not surprised by anything Baker does. This was his first opportunity, and he'll grow from it."
Taylor, expected to be Jackson's starter as Mayfield learns this season, hit all five of his passes in two series, including a 36-yard touchdown pass to Njoku, a New Jersey native.
Callaway, the Browns' fourth-round pick, played despite being cited Sunday for marijuana possession and driving with a suspended license. Police also found bullets and a gun part in his car during the stop.
He had three catches for 87 yards, including a diving 24-yarder the Browns successfully challenged after it was ruled incomplete on the field.
The Associated Press contributed to this report.