Playing in front of a limited number of fans for the first time this season, the Bills defense batted down Rivers' Hail Mary attempt as time expired to clinch a spot in the AFC divisional round.
Buffalo's last postseason win came on Dec. 30, 1995, when the Bills beat the Miami Dolphins 37-22 in the wild-card round. The Bills snapped a 0-6 playoff skid by winning Saturday.
"It's been a long time since Bills mafia has been able to celebrate like this," Bills star quarterback Josh Allen said after the win. "But again, it's one game. I am just happy for a chance to be able to play next week."
The Colts grabbed a 10-7 lead midway through the second quarter and almost extended it when Rivers slightly overthrew rookie receiver Michael Pittman Jr. in the end zone on fourth down.
After the fourth-down stop, the Bills scored 17 unanswered points to take a 24-10 lead early in the final frame. Rivers tossed two touchdown passes in the fourth quarter to trim the Colts' deficit to 27-24.
Trailing by three points with 2:30 left in the game, the Colts marched down the field and converted a fourth-and-10 after Rivers completed a pass to receiver Zach Pascal, who appeared to fumble after the catch. Officials ruled that Pascal was down by contact, and the call stood after a booth review.
The controversial sequence gave the Colts one final chance to force overtime or win the game. The Bills then knocked down Rivers' Hail Mary attempt to clinch the win.
The 24-year-old Allen completed 26 of 35 passes for 324 yards and two touchdowns in the victory, adding 54 yards and a score on the ground. According to ESPN Stats & Information, he became the youngest player in NFL history to record 300 passing yards and complete 70% of his passes in a playoff game.
Bills receiver Stefon Diggs had six receptions for 128 yards and a touchdown. Tight end Dawson Knox had two catches for five yards and a receiving score.
Rivers was 27 of 46 passing for 309 yards and two touchdowns. Rookie tailback Jonathan Taylor recorded 78 rushing yards and one score on 21 carries.
The Bills will now host a divisional playoff game next weekend, with their opponent to be determined.
"One's not good enough for us," Allen said. "We've got to find a way to put our best foot forward this week and get another one."