Jones, who led Ohio State to the 2015 national championship, threw a 4-yard touchdown pass to Jarrett Boykin as time expired.
But Jones failed to pull out the win by throwing an incompletion while attempting a 2-point conversion.
Tolzien, an offseason free-agent addition, shook off a slow start to produce two scoring drives in the first half. And then Morris, who was signed off Philadelphia's practice squad in December, secured the win with a 57-yard touchdown pass to Chester Rogers with 10:42 remaining.
First he caused a safety by blocking Pat McAfee's punt, which the punter swiped out of the end zone 6½ minutes in. On Buffalo's ensuing possession, Gragg caught a perfectly placed 19-yard fade pass from E.J. Manuel. Kicker Dan Carpenter hit a 37-yard field goal, but also missed a 53-yard attempt wide left in the fourth quarter.
Indianapolis had several reasons to not take any risks starting Luck. The field was slick after a pregame downpour. And there was also the prospect of the Colts revamped offensive line facing Rex Ryan's attacking defense.
Before the game, Colts owner Jim Irsay sent two tweets saying the team was holding out Luck until they returned to Indianapolis to play back-to-back home games. In the second post, he mentioned the rain, while also noting "my good friend Rex" isn't known for letting up in the preseason.
The Colts are taking no chances after Luck missed the final seven games of last season with a lacerated kidney and a partially torn abdominal muscle. And Irsay invested heavily in Luck by signing him to a six-year $140 million contract extension last month.
Copyright 2016 by The Associated Press