The La Liga side had to play the final quarter of an hour with 10 men after they used all three substitutes and the injured Aldo Duscher was unable to continue.
They had looked to be cruising to a fourth win out of four in the group when they carved open Stuttgart's defence with a slick passing move in the 14th minute.
Ndri Romaric picked out Luis Fabiano and the Brazilian striker's first-time through ball released Jesus Navas to finish coolly past the onrushing Jens Lehmann.
Sevilla goalkeeper Javi Varas did well to sprint off his line and smother an effort from Arthur Boka from a tight angle in the 36th minute but Stuttgart offered little else in attack before the break.
They showed more fight in the second half after a stiff talking to from coach Markus Babbel and a couple of changes.
Substitute Sebastien Rudy just failed to turn in a floated free kick in the 52nd minute and Pavel Pogrebnyak nodded against the crossbar two minutes later.
The dynamic of the match changed in the 74th minute when Duscher, who had only been on the pitch for 12 minutes, had to go off and the Bundesliga side pressed forward for an equaliser.
Zdravko Kuzmanovic found it when he picked up the ball in a central position just outside the area in the 79th minute and arrowed a shot into the top corner from 20 metres.
"I am a bit upset because we could have sealed first place in the group tonight," Sevilla coach Manolo Jimenez said at a news conference. "We played a decent first half against a very good team," he added.
"Only after Duscher was injured were they able to dominate."
Sevilla had Unirea Urziceni's Marius Onofras to thank for making it through to the next round after he scored a late equaliser in a 1-1 draw against visiting Rangers.
Sevilla have 10 points from four matches, Unirea have five, Stuttgart three and Rangers two.
