During his press conference earlier on Monday, Koetter said he had decided on a starting quarterback but was not ready to release that information to the public before talking with the two quarterbacks.
In his past three starts, Fitzpatrick has thrown seven interceptions and lost a fumble in three games.
While starting the first three games of the season while Winston was sitting out a three-game suspension, Fitzpatrick completed 70.3 percent of his passes for 11 touchdowns and four interceptions while throwing for more than 400 yards each time.
He started one more game before Winston took over as the starter.
After entering Sunday's game, Winston completed 12-of-16 passes for 199 yards and two touchdowns. Winston led four touchdown drives that helped the Buccaneers get back into what had been a one-sided game.
Koetter described Winston's play on Sunday as "fantastic."
Winston has had turnover issues this season. In fact, that is what caused him to be replaced as the starter.
In five games this season, including three starts, Winston has thrown eight touchdown passes and 11 interceptions, and he has also lost one fumble.
The Buccaneers have lost seven of their past eight games to drop to 3-7 on the season.