Stuttgart deserved their win before the 24,000 crowd but the two goals Boro scored in Germany earlier this month put the Premiership side through to the next stage.
Middlesbrough will now play Roma, with the first leg at the Riverside Stadium.
Manager Steve McClaren had fired a warning in his pre-match column.
"Stuttgart are a premiere side who don't concede many goals and they will fight every inch to claw back our advantage tonight. I would not expect anything else from a German team," he wrote.
"I said before the last draw that I wanted to face a big club and Stuttgart have much more experience in Europe than we have."
McClaren left leading scorer Yakubu out of his side and also put fit again Mark Viduka on the subs bench.
The Boro boss decided to pack his midfield, with veteran Jimmy Floyd Hasselbaink as the lone striker.
Stuttgart have a reputation for playing better away from home and they went to work from the kick-off.
The German side were cleverer on the ball and in their movements and certainly looked more dangerous in the opening stages.
Stuttgart overran the home midfield and it was no surprise when they took an early lead through Christian Tiffert, with his shot giving Mark Schwarzer no chance.
The goal certainly encouraged the visitors and the Middlesbrough defence looked shaky under pressure.
In contrast, the hosts struggled to show any threat in attack with Hasselbaink having limited service.
Boro were happy to be only a goal down at the interval after an indifferent display and there was some booing from a section of the crowd as they left the field.
It was expected that McClaren would introduce Yakubu or Viduka after the break to give Hasselbaink more support, but he opted to keep his strikers on the bench and defend Boro's away goals advantage.
Hasselbaink missed a great chance in the opening minute of the second half, even though he looked in an offside position as he darted forward, and with only Timo Hildebrand to beat he shot straight at he goalkeeper.
A goal at that stage would certainty have calmed the crowd.
Tiffert then shot past the far post shortly after being booked for a foul on Stewart Downing.
The crowd roared in anticipation as Gaizka Mendieta attempted to meet a Hasselbaink cross from the right, but the former Lazio player just failed to connect, though in any event, the flag was raised for offside.
Mendieta also saw an effort cleared off the line by Markus Babbel, but it mattered little as Boro hung on to book an encounter with the Serie A giants.