Mallett threw for 228 yards and a touchdown to lead the Texans to the 19-9 victory over Tampa Bay.
''We have to play better complementary football,'' coach Bill O'Brien said. ''We're going to work hard to do that, but I really liked the approach, the mental toughness that our guys had today.''
Mallett gave Houston (1-2) an early lead with a 5-yard pass to DeAndre Hopkins on the opening drive.
Mallett got his second NFL win in his second start this season after Brian Hoyer was benched following a lackluster performance in Houston's opener.
''It feels good,'' Mallett said. ''But we need more, we want more.''
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. 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.
Top overall pick Jameis Winston threw for 261 yards with a touchdown and an interception for Tampa Bay.
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 J.J. 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.
''We're working to find our brand and get back to playing our brand of football,'' Watt said. ''I still wouldn't say we're a hundred percent there yet, but I think today was a great step in the right direction toward where we want to be.''
Some things to know about Houston's win over Tampa Bay.
BLUE GOES FOR 100: Blue ran 31 times for 139 yards on Sunday for Houston's first 100-yard rushing game this season. The second-year player started the third straight game in place of Arian Foster, who is recovering from groin surgery. It was the second 100-yard game of his career, and his touchdown was the third rushing score of his career.
EVANS GETS GOING: Tampa Bay receiver Mike Evans had seven receptions for 101 yards on Sunday in his first unrestricted action of the season. Evans, the seventh overall pick in 2014, made his season debut last week after sitting out the first game with a hamstring injury in Week 1. He didn't catch a pass last week in limited playing time, but was Winston's go-to target on Sunday. ''When you have a great receiver like that, you have to try and get him the ball,'' Winston said. Evans dropped a few passes and admitted to a little rust after missing time with his injury.
JOSEPH'S BIG DAY: Veteran Houston cornerback Johnathan Joseph tied a career high with five passes defensed. The last time he broke up that many passes was on Nov. 8, 2009. He also had two tackles on Sunday. ''Anytime you have a quarterback that can make all the throws, it is going to be a tough day,'' Joseph said. ''I think we held up in coverage and we competed all day.''
STANDING BY BRINDZA: The Buccaneers were quick to voice support for Brindza despite his struggles on Sunday. Coach Lovie Smith was asked if he would remain the team's kicker.
''We are not going to indict a player based on one bad game that he had,'' Smith said. ''It happens, simple as that. He'll come back next week. He hit one long good one to get us off, a good start right away. That won't happen again.''
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