The hosts had taken the lead through Claudio Pizarro, however Yuya Osako had a goal disallowed before Mario Gómez scored twice just before half-time, with one thanks to a howler from Timo Horn.
Andreas Beck then scored to put the game beyond them, despite Milos Jojic scoring late on from a free-kick.
Pizarro gives Köln perfect start
After their impressive win at RB Leipzig last weekend, Stefan Ruthenbeck made just the one alteration, with the ill Jannes Horn replaced by Pizarro for just his second start for the Billy Goats. Stuttgart, with ten points from four games under Korkut, were unchanged following their 1-0 victory against Eintracht Franfkurt.
Köln were looking to move off the bottom for the first time since September with a win, and they got off to a dream start. Osako spied Pizarro on his left after Marco Höger had dispossessed Emiliano Insúa. The Peruvian rolled back the years by striking past Ron-Robert Zieler, to score his first goal for the club, having been denied in previous games by VAR.
Gómez and Eric Thommy had opportunities for Stuttgart, but otherwise they were struggling to get a grip on the game, constantly giving away the ball with Köln dictating the play. Pizarro had another chance to score when he headed wide a Jonas Hector cross, whilst Vincent Koziello robbed Holger Badstuber but after his ball found Simon Terodde, the striker rushed his shot and hit it wide, ruing the missed chance to score against his former club.
Things quickly turn sour for Billy Goats
Köln then doubled their lead, only for public enemy number one in these parts to strike again. Terodde was floored in the box but managed to flick the ball towards Osako. Zieler got to it though, however he failed to control it and with one hand on the ball he was disposed by Koziello, with Osako netting. Referee Sören Storks went straight to VAR though, perhaps without even making an initial decision, and a few second later the goal was disallowed.
That, plus what transpired in the final minutes of the half, summed up their season to a tee. Beck and Thommy combined on the right for Stuttgart, and although Höger nearly got the ball of him, he slipped which allowed Thommy to cross. He found Gómez, who fired in to equalise with the half-time break looming.
It would get even worse for the home side before that though. Gómez on the edge of the box tries to hook the ball round, hitting it though straight to Timo Horn. Normally he would have taken it with ease, but inexplicably he let it slip, with Frederik Sörensen unable to get back in time to spare his blushes.
Beck third goal proves cruical
Fortunately they had a chance to recompose themselves, but it didn’t make much difference as Stuttgart piled on the misery. Thommy’s influence on Stuttgart’s improvement in this game continued, as he passed to Beck. He then struck through a crowded box to beat Horn on the far side of the net to make it 3-1, his first Bundesliga goal in over four years.
There was a response of sorts from Köln, with Koziello, who was impressing again on just his second start, having his shot saved by Zieler after Terodde laid off, however otherwise they looked pretty well beaten. Stuttgart couldn’t take full advantage though, with a Gómez effort saved by Horn before Thommy put the ball over the bar.
Leonardo Bittencourt and Hector then both went wide for Köln, but they were given a glimmer of hope after Hector was fouled just on the edge of the box by Timo Baumgartl. Jojic, only a few minutes after coming on, hit the free-kick perfectly, over the wall and beating Zieler, who had been remonstrating with his defenders moments before.
Remarkably Baumgartl then gave Jojic another chance with a set piece, from a similar position but a little further out. This time he could only hit the wall though, and Köln failed to create any more chances in the four minutes of stoppage time. A big chance to claw their way towards possible survival has been missed, with just nine games now left to play and time running out. Stuttgart, on the other hand, are up to ninth.