"I think that you guys all feel good about where things are headed, so let's just wait and see and go from there," Jackson said on Friday.
Mayfield, who was selected with the top overall pick of the 2018 NFL Draft, completed 17-of-23 passes for 201 yards to help the Browns overcome a 14-point deficit and record their first win since Dec. 24, 2016.
Mayfield led the Browns to a field goal on his first possession to make it a 14-3 deficit at halftime. He guided Cleveland to another field goal in the third quarter before directing two lengthy touchdown drives that were capped by a pair of 1-yard runs from running back Carlos Hyde.
Taylor was 4-of-14 for 19 yards before leaving late in the first half on Thursday. He is 41-of-84 for 462 yards, two touchdowns and two interceptions this season.
Acquired in an offseason trade with the Buffalo Bills, Taylor is completing only 48.8 percent of his passes and owns a quarterback rating of 63.7.
The Browns have used 29 different starting quarterbacks since the franchise relocated back to Cleveland in 1999. The player who almost assuredly will be No. 30 said the city should get used to more success.
"Cleveland deserves a win, but we're not done yet," Mayfield said. "So don't break the dilly dilly coolers too hard; just enjoy it. We deserve it, but at the same time we're just getting started."