With Unirea Urziceni winning 4-1 at Rangers, Stuttgart are now in the bottom two of Group G and have only two points from three games and face a mountain to climb if they are to survive the group stages.
In contrast, Sevilla have nine points from three games and are virtually assured of progressing after they raced into a 3-0 lead thanks to two headers from French defender Sebastien Squillaci and midfielder Jesus Navas.
Stuttgart pulled a consolation goal back through Brazil's Elson, but for the Swabian side it was too little, too late.
"We are making too many simple mistakes at the moment," said Stuttgart coach Markus Babbel. "But today the performance and the result do not fit. However, I am confident, because I have seen a good performance from us here today."
Sevilla took the lead against the run of play when their Brazilian midfielder Adriano swung in a cross and Squillaci netted, looping his header over diving Stuttgart goalkeeper Jens Lehmann on 23 minutes.
It was extremely harsh luck for Stuttgart whose Belarusian play-maker Alexander Hleb had been causing havoc in the Spanish defence and the Germans had enjoyed the better of the opening exchanges.
Stuttgart's Brazil striker Cacau fired straight at the goalkeeper with an earlier effort while Serbian midfielder Zdravko Kuzmanovic scored only to have his effort ruled offside.
Sevilla made sure of the three points when Argentina midfielder Diego Perotti beat the defence and chipped in a cross which Lehmann failed to gather and midfielder Jesus Navas fired home from close range on 55 minutes.
Squillaci scored with a header when Serbia midfielder Ivica Dragutinovic curled in a free-kick and with a hint of offside the Frenchman beat Lehmann for the second time on 72 minutes.
Stuttgart pulled a deserved goal back when Elson curled in a direct free-kick on 74 minutes from 25 metres out.
But it was no more than mere consolation as the Spaniards left Germany with another three points under their belt.