The midfielder fired in five minutes from time at the Stade de France following goals from Luis Muriel and Radamel Falcao, after the home side had led through Olivier Giroud and Thomas Lemar.
France started with Paul Pogba on the bench but named an otherwise attacking line-up, with Lemar, Kylian Mbappe, Antoine Griezmann and Giroud in attack, and the Chelsea striker broke the deadlock with his 30th goal for his country after an error from David Ospina.
Lemar blasted home a second at the end of a sublime counter-attack, but Colombia hit back two minutes later, with Muriel's cross curling into the far corner of the net.
The goal seemed to shock the hosts, who suddenly looked uncertain on the ball and survived two close calls through Muriel before Falcao buried his side's second just past the hour mark.
Samuel Umtiti's lunge on Jose Izquierdo allowed Quintero to complete a famous comeback, leaving France with plenty to ponder before they visit World Cup hosts Russia in their next match on Tuesday.
The speed of France's wide players caused plenty of problems for Colombia early on, and it was no surprise to see the opening goal come from an overlapping run down the left from Lucas Digne.
The Barcelona full-back latched onto Lemar's quite brilliant throughball, before sending in a low cross that Ospina fumbled into the path of ex-Arsenal team-mate Giroud, who gratefully fired high into the net.
Lemar doubled the lead 15 minutes later at the end of a delightful counter-attack. Griezmann backheeled Djibril Sidibe's path to Mbappe, who skipped into space and teed up his old Monaco team-mate to rifle into the right-hand corner from inside the penalty area.
Colombia hit back barely two minutes later, though, as Muriel's cross from the left of the France box curled into the corner, with Hugo Lloris wrong-footed by Davinson Sanchez's attempt to reach the ball.
Griezmann was denied by a good Ospina block but the goal seemed to have unsettled the home side, and Muriel missed a great chance three minutes into the second half after being picked out in space by Carlos Sanchez.
The Sevilla striker then robbed Raphael Varane only for Lloris to make a good save, as Didier Deschamps became increasingly animated on the touchline, evidently perturbed by his side's second-half performance.
His shouts did not have the desired effect. With 62 minutes played, N'Golo Kante cheaply conceded possession 30 yards from the French goal, Carlos Sanchez fed James Rodriguez and his cut-back was swept home by Falcao, who was unmarked 12 yards out.
Pogba – winning his 50th cap, like Griezmann – and debutant Wissam Ben Yedder were promptly introduced, and the former took little time to make an impact, stinging Ospina's palms with a dipping effort from 25 yards out.
There was also a first appearance for Lucas Hernandez, on for the injured Digne, but it was Quintero who proved to be the hero, dispatching his spot-kick with aplomb after Umtiti was harshly penalised for a challenge on Izquierdo, despite having made contact with the ball.