O'Brien benched starting quarterback Brock Osweiler midway through the second quarter on Sunday, with Houston trailing the Jacksonville Jaguars 13-0. Savage led the Texans to a comeback 21-20 victory in relief of Osweiler.
O'Brien said he thinks Savage gives the Texans the best chance to beat the Bengals.
After the game on Sunday, O'Brien said, "We don't make decisions on how much a guy gets paid. We make decisions on what's the best way to win a game."
The Texans gave Osweiler a four-year, $72 million contract in March, though just two years and $37 million of it is guaranteed. Savage, who is in the third year of his rookie contract he signed after being drafted by the Texans in 2014, is guaranteed just $300,584 for the remainder of his deal.
Savage finished 23-of-36 passing for 260 yards after entering in the second quarter on Sunday. Before he was benched, Osweiler was 6-of-11 for 48 yards and two interceptions. Neither quarterback threw a touchdown pass.
O'Brien said he chose to name a starter early in the week because he wanted to give Savage the full week to prepare during what is already a short week.
"I think it's important for Tom to have the week as a starter, to experience the whole thing and to make sure he understands what the game plan is," O'Brien said. "It's a short week, so he knows he's got to put a lot of extra prep time into this thing, and I know he will."
Prior to Sunday, Savage had not played in a regular-season NFL game in more than two years. He spent last season on injured reserve after spraining his shoulder during a preseason game. During training camp, O'Brien praised Savage's knowledge of the offense.
"He's the same guy every day," O'Brien said. "He's a very consistent person. He's been the same guy as far as his personality and how he approaches his day-to-day workload since he arrived here. He's a hardworking guy. He's done a lot to work in the weight room to change his body. He's worked to get in better shape over the years.
"He's always trying to do something to prepare, and I think as far as leadership in the huddle, he communicates the play well. He can throw the football, and that more than anything can spark a team ... when you complete some passes."
Savage will be the sixth different starting quarterback for the Texans in the past two seasons. Only two other teams have had five different starters the past two seasons, according to ESPN Stats & Information research.
The Texans are in a playoff race, tied with the Tennessee Titans atop the AFC South at 8-6. Houston holds the tiebreaker since the Texans are 5-0 in the division. The Texans and Titans play Week 17 in Tennessee.