Leading from the first leg and going into the game on the back of a superb victory against Manchester United, Everton were widely expected to complete the job in Portugal.
However, Sporting shrugged off their indifferent form to progress to the last 16 thanks to second half goals from Miguel Veloso, Pedro Mendes and substitute Matias Fernandez.
Everton had been thrashed 5-0 by Benfica on their last visit to Lisbon in October but David Moyes was able to name a strong side for this occasion and better was expected.
However, they were on the back foot from the very first whistle and Tim Howard was forced into action after just four minutes when Marat Izmailov cut inside and cracked a shot the American parried to safety.
It was an early warning to Everton that Sporting were determined to turn this tie around and there was another reminder in the 13th minute when Liedson clipped his effort wide of the post.
There was to be an even bigger let-off for the Toffees soon after, however, when Phillipe Senderos barged over Yannick Djalo on the edge of the area and Joao Moutinho send a sweet free-kick crashing onto the bar.
Liedson latched on to the rebound but snatched at his effort and it went wide.
Veloso sent a shot into Howard's body as Sporting maintained the momentum but Everton woke from their slumber a minute before half-time when Senderos found the net after getting on the end of a free-kick from Leighton Baines. His effort was ruled out for offside.
Senderos needed treatment to his back three minutes into the second half after taking a hefty knock and he was eventually replaced in the 52nd minute by Phil Jagielka, making his first appearance since April.
Djalosent a shot narrowly over the bar shortly before that and Jagielka, drafted in without any reserve games under his belt following his knee ligament injury, must have known his side were up against it.
Howard did well to turn away a close range shot from Izmailov in the 58th minute and he showed even better reflexes to turn over a rasping drive from Veloso shortly after.
Steven Pienaar saw his effort go narrowly over the bar as Everton fleetingly threatened, but Sporting made the breakthrough they deserved in the 64th minute when Veloso got clear inside the six yard area and drove past Howard from a tight angle.
That made it 2-2 on aggregate but things got worse for the Toffees in the 76th minute as Mendes got on the end of a cutback from Izmailov to find the corner with his effort, via a deflection off substitute Jack Rodwell.
Everton pushed forward in the dying minutes but a Sporting breakaway in stoppage time saw Fernandez net after he fortuitously stumbled around Howard.