Chiefs tight end Kelce produced a vintage performance in tandem with quarterback Patrick Mahomes (29) to overpower a stubborn Texans lineup.
Kelce finished with 117 receiving yards from seven receptions while Mahomes threw for one touchdown with 177 passing yards.
"You know 87's gonna show up whenever there's a big-time moment and he did that," Mahomes said of Kelce. "Everybody's been asking 'Where's Travis Kelce at?' I think he showed the world where he's at."
Meanwhile, the Chiefs' defence delivered another ferocious performance, terrorising Texans quarterback C.J.Stroud, who was sacked eight times in a torrid outing.
The divisional round victory at Arrowhead Stadium sends Kansas City into the AFC Championship game for a seventh straight season, where they will host either the Baltimore Ravens or the Buffalo Bills.
Saturday's victory also marked a milestone 300th career win for Chiefs head coach Andy Reid - only the fourth head coach in NFL history to reach that mark.
The Chiefs, who clinched back-to-back Super Bowls with their victory over San Francisco last year, aim to become the first team in history to win three straight Super Bowls.
The Chiefs got off to a dream start when receiver Nikko Remigio weaved his way deep into Texans territory from the kick-off with a 63-yard return.
The Chiefs were unable to cash in though and were forced to settle for a Harrison Butker field goal from 32 yards to take a 3-0 lead.
Texans kicker Ka'imi Fairbairn levelled with a field goal on the next possession, before Butker slotted a 36-yard field goal to put the Chiefs 6-3 up at the end of the first quarter.
The familiar Kelce-Mahomes double act then sparked the Chiefs to the opening touchdown in the second quarter.
A short pass up the middle from Mahomes found Kelce who spun clear of the Texans defenders and surged away for a 49-yard gain that left Kansas City camped on the Houston six-yard line.
That led to a Kareem Hunt rushing touchdown from close range to put the Chiefs 13-3 up, before Fairbairn cut the deficit just before half-time with his second field goal.
The Texans should have drawn level in the third quarter after quarterback Stroud orchestrated a superb 15-play, 82-yard drive that burned more than 10 minutes off the clock and ended with running back Joe Mixon scampering into the end zone from 13 yards.
But Fairbairn missed the extra point to leave the Chiefs 13-12 ahead, and Mahomes made them pay on the next possession, finding Kelce in the end zone while off balance to help Kansas City take a 20-12 lead.
Houston were unable to score from their next two possessions and the Chiefs went 11 points clear with Butker's third field goal late in the fourth quarter. With 11 seconds remaining, the Chiefs conceded a deliberate safety to surrender possession and effectively ice the game.