Chiefs quarterback Mahomes finished with 321 passing yards from 23 of 35 attempts at Arrowhead Stadium to send the Chiefs into a home AFC Championship showdown against the Tennesee Titans next weekend for a place in the Super Bowl.
"We didn't start the way we wanted to," Mahomes said. "All we were preaching was, 'Let's do something special.
"'Everyone is counting us out, let's just keep fighting, one play at a time.'"
Texans head coach Bill O'Brien said his team had not taken anything for granted after racing into an early 24-0 lead.
"24-0 was a really good lead, no doubt about it," O'Brien said.
"But I don't think any lead is safe versus these guys. We needed to score 50 points and we weren't able to do that."
Texans quarterback Deshaun Watson, meanwhile, finished with 388 yards and two touchdowns in a losing effort.
The Chiefs were given an almighty scare after a disastrous start.
But with the Chiefs reeling, the Texans gave the hosts respite from the onslaught early in the second quarter with Houston coach Bill O'Brien opting for the safety-first option of a field goal on fourth down and 1 from 31 yards.
Ka'imi Fairbairn drilled the kick to put Houston into a 24-0 lead.
But momentum shifted on the ensuing kickoff when Mecole Hardman returned 58 yards to the Houston 42-yard line.
Two plays later, Mahomes hit Damien Williams from 17 yards to make it 24-7 and put the Chiefs on the board.
Kansas City were then gifted another opening after a risky decision by O'Brien to fake punt on fourth down inside the Texans half.
The move backfired and the Chiefs took over possession, with Mahomes finding tight end Travis Kelce from five yards for a touchdown.
A disastrous spell for the Texans then saw DeAndre Carter fumble the kickoff return, allowing the Chiefs to recover at the Houston six.
Three plays later, Mahomes hit Kelce for the tight end's second touchdown to close the gap to 24-21.
The blitz continued after Houston punted away their next possession, Mahomes marching the Chiefs 90 yards upfield before flipping a short pass to Kelce into the end zone to hand them a 28-24 lead at halftime.
The scoring continued through the second half, with Williams adding two rushing touchdown to put the Chiefs 41-24 ahead.
Watson rushed for a five-yard touchdown to put the Texans back to within 10 points but the Chiefs added another touchdown followed by a field goal to extend their lead at 51-21.
Texans coach O'Brien admitted the risky attempted fake punt in the second quarter had proven costly.
"We had momentum at that point and it felt like we were going to try and make a play," O'Brien said. "It just didn't work."