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The Bengals converted a fourth down play to keep a drive alive, but four plays later, former Texan Randy Bullock's 43-yard attempt sailed wide right to secure Houston's victory.
Savage helped the Texans (9-6) rally for a win over the Jaguars last week after taking over when Brock Osweiler was benched after throwing interceptions on back-to-back possessions in the second quarter.
The Texans knew they were in the postseason with a win after the Titans fell to the Jaguars earlier Saturday. But it didn't look good for them early with Savage completing just two passes for 13 yards and them trailing 3-0 at halftime.
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Savage was able to move the offense much better in the second half, but Houston still struggled in the red zone.
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The Texans had a first down at the 6 early in the fourth quarter before the offense stalled and they settled for a 22-yard field goal that put them up 6-3. Quintin Demps got his career-high fifth interception when he picked off Andy Dalton's pass two plays later, but Houston couldn't move the ball and had to punt it away.
it 86-yards for a touchdown that made it 10-6 with about 11 minutes left.
A 24-yard touchdown run by Blue put the Texans back on top with about 9 minutes left, but the extra point was blocked to leave Houston up 12-10. That score was set up by a 21-yard reception by DeAndre Hopkins.
Dalton finished with 268 yards passing with a touchdown and an interception.
Lamar Miller missed his first game this season because of an ankle injury and Blue had 73 yards rushing in his place.
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A.J. Green, the Cincinnati star receiver who said earlier this week that he would play after missing the last four games with a hamstring injury, was inactive amid reports that he was unhappy with the decision to sit him.
After a terrible first half on offense, Houston finally got a little something going after halftime and tied it at 3-3 with a 25-yard field goal early in the third quarter.
The first half was an offensive struggle for both teams with a combined 12 punts. The Texans were a bit worse than the Bengals though and had just two first downs and 34 total yards with minus-5 yards passing because of three sacks.
The Bengals had a 3-0 lead at halftime after a 43-yard field goal by Randy Bullock at the end of the second quarter.
The Texans didn't cross midfield their first drive of the third quarter when Will Fuller caught a 14-yard pass that got them to the Cincinnati 47.
The Associated Press contributed to this report.