City, trailing 1-0 from the first leg, came from 2-0 down at half-time to win on the night, with goalkeeper Joe Hart denied an incredible winner in the final second by a brilliant save from Rui Patricio.
Sergio Aguero scored twice and won a penalty, converted by Mario Balotelli, as the home side hit back from goals by Matias Fernandez and Ricky van Wolfswinkel.
Roberto Mancini's side threw the kitchen sink at the Portuguese outfit late on, with Hart coming up for a corner in the last minute of injury time only to see his goal-bound header diverted just wide.
City's exit came two hours after local rivals and fellow Premier League title chasers Manchester United went out to Athletic Bilbao, meaning there are no English sides left in the Europa League with only Chelsea remaining in European competition.
City were booed off after a rotten first half in which they barely tested Sporting keeper Patricio and conceded two goals from individual errors.
They had been warned too - Xandao heading a corner wide, and Fernandez booked for diving when a less theatrical reaction may have seen his Chile team-mate David Pizarro concede a penalty.
The hosts were as poor going forward as they were defensively, Balotelli the main culprit as he fell over when put clean through, before delaying his shot after again being found in space, allowing defender Anderson Polga to block.
Balotelli veers from the sublime to the ridiculous so it was perhaps fitting that his erratic behaviour would lead to Sporting’s opener on 34 minutes.
On the edge of his own box the Italian - blatantly and needlessly - shoved Emiliano Insua: from the resultant free-kick Fernandez curled a beauty around the wall and inside the far post, the pace too much for Hart, who was perhaps a shade too far off his line and could have done better.
City immediately looked for a response as Aguero almost met a fine David Silva cross.
But they soon conceded another when - after a poor clearance from Aleksandr Kolarov and the failure of Silva to track back - Marat Izmailov picked out Van Wolfswinkel, who tapped his low cross into the roof of the net from a couple of yards out.
The jeers soon came - the home fans spoilt by recent success but hardly helping the cause of a side fighting for their first league title in the Premier League era - with Mancini a dejected figure as his troops trudged off.
Mancini’s half-time change did not seem positive - Adam Johnson, reasonably bright, was replaced by Nigel de Jong - but City were a lot more solid, while Yaya Toure was given a more advanced role.
And after 15 minutes of pressure City were back in it: on the hour mark Toure slipped a delightful pass into the space between the centre-backs, with Aguero controlling with one touch and rattling a vicious finish past Patricio with his second.
City still needed three more and they had a real go at the Portuguese, throwing on Edin Dzeko and Samir Nasri and putting extreme pressure on the visitors’ goal.
They pulled back a second soon afterwards, levelling on the night when Aguero was tripped by sliding substitute Renato Neto just inside the penalty area: Balotelli sent an impudent rolled finish past Patricio, meaning two goals were needed in 15 minutes.
The first came with seven minutes of regular time left as, after Stefan Savic flicked on a Kolarov corner, Aguero was left unmarked at the far post, the Argentine guiding a cool volley past Patricio.
Sporting resorted to some extreme time-wasting, which Norwegian referee Tom Harald Hagen acknowledged when he added on five minutes at the end.
City were piling everything forward and they had their chances too: Dzeko should have done better when heading wide as Kolarov, now operating as a winger, rained in one of many crosses from the left.
Balotelli thought he had scored from an identical situation, but Anderson Polga’s faint touch sent his header wide, while right at the death - and after some more comic gamesmanship from Sporting - there was almost a fairytale ending.
In the fifth and final minute of injury time Kolarov’s initial corner was cleared - but as the second ball came in Hart, who had come up for the set piece, remained in the box, escaping the attentions of Insua to power a header on target: Patricio got a faint touch, sending the ball shaving the post and wide, while the referee blew time, sending City out of Europe for a second time this season.
It was a valiant effort but, for all their spending, City are now out of all the cups and lie second in the Premier League: a season without silverware is a realistic possibility, with success and failure set to hinge on the Manchester derby on April 30.