Patriots running back Danny Woodhead went out early with a thumb injury and stand-out tight end Rob Gronkowski re-broke his left forearm when he hit the ground trying to make a catch in the first quarter.
Brady, pulling the strings of the NFL's most potent offense, struggled in the first two drives before figuring out how to unlock the Texans' defense and leading New England to a 41-28 victory that put the Patriots one win away from the Super Bowl.
It was the 17th career playoff victory for Brady, moving him past boyhood idol Joe Montana to the top of the all-time list.
Brady said the game plan included a lot of options for Woodhead and Gronkowski, who was playing just his second game since previously breaking his forearm.
"We run the first series of the game and all those plans change," Brady told reporters. "I think there was a little bit of, 'What are we going to do now, how are we going to adjust?'"
It did not take long for Brady to figure it out and he found plenty of work for second-year running back Shane Vereen.
Vereen scored on a one-yard run in the first quarter, got into the end zone untouched after a swing pass from eight yards out in the second, and made a dazzling over the shoulder catch for a 33-yard touchdown to put the Patriots out of sight.
"We seemed to settle in there midway through the first quarter and put together a pretty good game," said Brady, who has won three Super Bowl rings and been runner-up twice more.
"Obviously it's a bummer to lose anybody but someone of Rob's importance or Danny's importance, we need guys to step in and fill the void."
Patriots coach Bill Belichick singled out back-up tight end Michael Hoomanawanui and Vereen for praise.
"Hooman and Shane stepped up and did a great job for us," the coach said. "Shane obviously made a lot of big plays.
"These kind of games you never really know when the dial spins, where it's going to end up, who it's going to end up on, and those guys were prepared and offensively we were able to move the ball put up some points."
Belichick said Brady had a lot to do with the seamless transition.
"We all follow him. We all respect him. He led the team today, along with a lot of other guys, but he certainly did his part, as he's done many times before," said Belichick.
"No quarterback I'd rather have than Tom Brady."
Belichick said Brady's understanding of defenses is the key.
"We used some different formations, tried to move guys around and as Tom usually does he finds the best match-ups depending on the route and the coverage and does a great job getting the ball to the guys that have a great opportunity to win on those routes," he said.
Brady threw three touchdown passes, completing 25-of-40 passes for 344 yards with no interceptions.
The Patriots host the Baltimore Ravens next Sunday with the AFC title and a trip to the Super Bowl on the line.
(Reporting by Larry Fine in New York, Editing by Peter Rutherford)