Mahomes steered the Chiefs back from a 20-9 fourth quarter deficit to force overtime before leading the decisive drive that culminated with Harrison Butker slotting a 27-yard field goal to clinch victory.
"This is exactly what we needed against a really good football team when the game's not going your way," Mahomes said afterwards.
"We could have folded in that situation (but) the guys responded and found a way to win. We needed a win like this - now let's try and build off it," added Mahomes, who finished with 352 passing yards with no touchdowns and an interception.
The Chiefs went into Sunday's game at Arrowhead Stadium knowing that a defeat could effectively mark the end of a dynasty, with the team outside of playoff contention with only six games remaining.
Instead, the four-time Super Bowl champions improved to 6-5 and now have a fighting chance of reaching the postseason for an 11th straight year - a remarkable run that has included five trips to the Super Bowl since the 2019-2020 campaign.
The Colts, who fell to 8-3 with the defeat, had looked poised to deal the Chiefs' season a terminal blow after controlling the game for long periods with quarterback Daniel Jones throwing two touchdowns before a pair of Michael Badgley field goals in the third quarter made it 20-9 to the visitors.
Once again, though, Mahomes answered the call when his team needed him most, orchestrating a 56-yard drive to set up a Kareem Hunt touchdown before making the two-point conversion to make it 20-17, and then moving his team 87 yards upfield for a game-tying Butker field goal as time ran out.
After the Colts were forced to punt in overtime, Mahomes then led an 81-yard drive for the winning field goal.
Cowboys stun Eagles
Elsewhere Sunday, the Dallas Cowboys pulled off an improbable comeback of their own in a thrilling 24-21 defeat of the reigning Super Bowl champion Philadelphia Eagles in Texas.
The latest instalment of one of the NFL's most ferocious rivalries served up a classic which saw Dallas recover from a 21-0 deficit to snatch victory with Brandon Aubrey's game-winning 42-yard field goal on the final play of the game.
The Cowboys' win saw them improve to 5-5-1 for the season and left the Texas club second in the NFC East behind the Eagles, who fell to 8-3.
In other games, the resurgent New England Patriots improved to 10-2 with a 26-20 win over the Cincinnati Bengals on the road to move closer to a return to the playoffs.
The Pittsburgh Steelers, meanwhile, slumped to a 31-28 defeat against the Chicago Bears at Soldier Field. The Bears improved to 8-3 thanks to Caleb Williams throwing three touchdowns in an impressive performance.
The Detroit Lions moved to 7-4 but were given a scare by the New York Giants before clinching a 34-27 win, while the Green Bay Packers downed the Minnesota Vikings 23-6.
The Seattle Seahawks remain top of the NFC West after bagging a 30-24 road win over the Tennessee Titans, while the Baltimore Ravens continued their return to form, clinching a fifth straight win in a 23-10 defeat of the New York Jets.
Cleveland Browns' rookie quarterback Shedeur Sanders won his first NFL start in a 24-10 defeat of the Las Vegas Raiders, while the Jacksonville Jaguars pipped the Arizona Cardinals 27-24 with an overtime field goal to improve to 7-4.