Anastasios Donis' first goal in Stuttgart colours was cancelled out in the second-half by Dominique Heintz's 20-yard strike and the game looked to be heading for a draw until late drama rewrote the script.
Substiute Sehrou Guirassy was awarded a penalty only for the VAR to overturn the decision in the 90th minute, before Chadrac Akolo netted in the 94th minute to send the Mercedes-Benz Arena into delirium and leave Köln without a win in eight.
Donis puts Swabians ahead with his first goal for the club
Hannes Wolf made three changes to the side that lost 2-1 at Eintracht Frankfurt prior to the international break as Akolo, Daniel Ginczek and Andreas Beck made way for Orel Mangala, Josip Brekalo and Simon Terodde.
Köln boss Peter Stöger meanwhile made just one change to the side that lost 2-1 at home to RB Leipzig on October 1, bringing in Simon Zoller for Jhon Córdoba to pair Yuya Asako up front.
Effzeh enjoyed the better of the opening 20 minutes but were unable to capitalise on the chances the created, with Osaka having two opportunities on goal only to send them wide, letting Stuttgart off the hook.
The hosts came into the game after their early scare and Donis should have put them 1-0 up on 32 minutes when Beck - who was brought on for the injured Marcin Kaminski - pulled the ball back to him only for the Greek forward to place his shot wide.
Stuttgart were somehow kept at bay only minutes later as a huge scramble in the Köln penalty area after Beck's cross led to several shots being fired in by almost every Stuttgart forward, but Timo Horn reacted well to block Donis' header at the near post before clearing.
Wolf's men found their breakthrough 38 minutes in as Köln were caught pushing too far forward by Mangala who won the ball back to play in the on-rushing Terodde, and the German picked the perfect moment to play Donis through and the on-loan Juventus man slotted through the legs of Horn to put his side ahead.
Köln see a point snatched from their grasp following late controversy
The visitors have scored just two goals in eight Bundesliga games this season, and so it was a surprise to see new signing Claudio Pizarro replace Salih Özcan at half-time rather than make the starting 11.
The veteran striker had a half-chance shortly after his introduction as the ball was whipped in to the back post, but he sent his header a few yards wide of the woodwork with Ron-Robert Zieler scrambling across goal.
The hosts were made to pay for their carelesness in the final third as Köln capitalised on Akolo's misplaced pass to race up the pitch, where defender Heintz cut inside and curled into the top corner of Zieler's net to equalise with just under 15 minutes remaining.
Effzeh thought they had been handed a golden opportunity to take the lead in the 88th minute as Guirassy was deemed to have been fouled in the area and a penalty was given, only for the VAR to overturn the decision after several minutes of deliberation.
Stuttgart capitalised on their stroke of good fortune to race down the other end and snatch all three points, as substitute Akolo cut inside the penalty area and saw his shot deflect off Köln defender Tim Handwerker and past the helpless Horn.