Rookie Kyle Brindza, who made a 58-yard field goal early in the second quarter, missed an extra point and three field goals in the second half.
"The field goal kicker kind of helped us out a little bit, so I appreciate that," Watt said, snickering.
Mallett gave Houston (1-2) a lead with a 5-yard pass to DeAndre Hopkins on the opening drive.
The Buccaneers (1-2) went up 9-7 on a touchdown reception by Charles Sims in the second quarter that came after an interception by Mallett, but didn't score again.
"We didn't block any of them," Watt said of the field goal misses. "They just didn't go through the yellow things ... sweet. I hope that happens every time."
Blue had a 20-yard touchdown run after Brindza's second missed field goal in the fourth quarter, but Randy Bullock missed the extra point to leave Houston up 16-9.
Mallett got his second NFL win in his second start this season after Brian Hoyer was benched after struggling in Houston's opener.
"Throughout my career, high school, college whatever, I always feel better the more I play, the more comfortable I get," Mallett said. "So hopefully I continue."
Coach Bill O'Brien saw improvement from Mallett.
"I think he understood how we wanted to manage the game better today," O'Brien said.
"Offensively, they established their run; we couldn't get ours going," Tampa Bay coach Lovie Smith said. "Dropped balls. We had opportunities that we didn't enough from, as simple as that."
Bullock added a 19-yard field goal with less than two minutes left.
Hopkins, who missed practice this week because of a concussion, finished with eight receptions for 101 yards.
Winston launched a 33-yard pass to Louis Murphy just before he was leveled by Watt early in the fourth quarter. He found Murphy again two plays later for a 19-yard gain, but the offense stalled and Tampa Bay was left empty-handed when Brindza missed from 33 yards.
On Houston's next drive Mallett connected with Hopkins for a 30-yard gain and the Texans got another 15 thanks to a facemask penalty. Blue, filling in for injured Arian Foster, capped that drive with his first touchdown this season.
"It was a play we practiced all week," Demps said. "You have to credit J.J. Watt and Clowney and those guys for getting pressure."
The Texans took a 10-9 lead on Bullock's 39-yarder.
Vincent Jackson grabbed a 23-yard reception that at first was ruled a touchdown in the third quarter. But the Texans challenged the call and it was reversed; Jackson didn't get both feet down. Brindza missed a 41-yard field goal.
Brindza's third miss on a 57-yarder late in the final period.
"He, like a lot of us, had a disappointing day today," Smith said. "You're going to have off days. This was an off day. When a player has an off day, we need to rally around him in other ways, and that's what we weren't able to do today."
Brindza lamented his tough day.
"It's always devastating and disappointing," he said. "I let my team down."
Watt, who had three touchdown receptions last season, lined up as a tight end for one play, but Houston ran the ball.
Mallett was sacked and fumbled near the end of the first quarter and Tampa Bay recovered. But the play was negated because of a defensive holding penalty on Jacquies Smith.
Tampa Bay cut the lead to 7-3 early in the second quarter on Brindza's 58-yard field goal.
The Texans were without LT Duane Brown (hand) for second straight game. ... Brindza's 58-yard field goal was second longest in franchise history. ... Tampa Bay LB Lavonte David had 14 tackles. ... Houston CB Johnathan Joseph tied career high with five passes defensed. ... The Texans changed to AstroTurf for Sunday's game after using natural grass in every game since the stadium opened in 2002.
AP NFL website: www.pro32.ap.org and http://twitter.com/AP
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