Baker Mayfield completed 25-of-38 passes for 215 yards and two touchdowns with no interceptions as the Bucs made a fifth consecutive 2-0 start to an NFL campaign while the Texans fell to 0-2.
Houston's Jaylin Noel returned a punt 53 yards to set up Nick Chubb's 25-yard touchdown run two plays later, but a missed two-point conversion pass left the Texans ahead by only 19-14 with 2:10 to go.
Mayfield sustained Tampa Bay's final drive with a 15-yard run on fourth down for a first down and directed an 11-play, 80-yard march capped by White's decisive run and a missed two-point conversion pass.
Mayfield threw a 20-yard touchdown pass to Ryan Miller and a 15-yard touchdown pass to Emeka Egbuka to give the Buccaneers a 14-10 halftime lead.
Tampa Bay's defenders stopped the Texans on downs at their one-yard line early in the fourth quarter.
But Houston's Jakob Johnson blocked a punt to set up Ka'imi Fairbairn's 53-yard field goal, pulling Houston within 14-13 with 5:22 remaining to set up the dramatic finish.
The Bucs stretched their lead atop the NFC South division while the Texans fell two games behind AFC South pace-setter Indianapolis.
At Las Vegas, the Chargers denied the NFL's oldest coach, the Raiders' Pete Carroll, a victory on his 74th birthday.
Herbert threw touchdown passes of 60 yards to Quentin Johnston and 10 yards to Keenan Allen on the way to a 17-6 halftime lead.
The teams exchanged second-half field goals and the Chargers' Donte Jackson intercepted a Geno Smith pass in the end zone to end a Las Vegas threat with 5:58 remaining.
The Raiders lost the ball twice more on downs and Los Angeles ran out the clock after that.
Burrow out for three months
Cincinnati quarterback Joe Burrow will undergo surgery on his left big toe, Bengals coach Zac Taylor said on Monday.
He is expected to miss at least three months with the "turf toe" injury, suffered when he was sacked in the second quarter of his team's 31-27 victory Sunday over Jacksonville.
Burrow, who was seen in a walking boot with a crutch after the game, missed the last six games in 2020 as a rookie with a knee injury and the last six games in 2023 due to a torn wrist ligament.
The Bengals are off to their first 2-0 start in seven years.
Burrow is not the only hurting NFL quarterback.
Minnesota's J.J. McCarthy has a high right ankle sprain and will miss next Sunday's game with the Bengals. McCarthy missed all of last season, what would have been his rookie campaign, with a right knee injury suffered in the pre-season.
Washington's Jayden Daniels suffered a sprained left knee in the Commanders' loss last Thursday at Green Bay and is questionable for next Sunday's game against Las Vegas, Washington coach Dan Quinn said.