With England overcoming Ukraine 1-0 in Donetsk, the French needed to match that scoreline to win Group D but failed miserably as Sweden - despite already being eliminated having lost their first two games - were by far the better side at the Olympic Stadium.
AC Milan forward Ibrahimovic set them on the road to three points when he swivelled to smash a volley low into the corner of the net from just inside the opposition area to earn Sweden the lead from Sebastian Larsson's cross on 54 minutes.
It was a special finish from the towering forward in a commendable team effort by Sweden, an asset that seemed to seep from the French side on such an important evening.
After continuing to trouble a France side that looked worryingly one-paced and were largely restricted to shots from distance, Sweden gleaned a merited second goal in added time.
Substitute Samuel Holmen knocked an effort off the bar, but Larsson had the presence of mind to whack the rebound into the gaping net with the French defence nowhere to be seen.
France were finished for the night, but of course are not finished for the tournament.
They concluded on four points - three points behind Group D winners England - but their reward for this insipid performance is a match against Group C winners Spain in Donetsk on Saturday night with England meeting Italy 24 hours later in Kiev.
Sweden depart the tournament with a first competitive win over France since 1969 - a run stretching back 12 games - while France have a few days to reflect upon losing their 23-match unbeaten record.
Philippe Mexes was harshly booked in the second half for a foul on Ola Toivonen, and will miss the match against Spain. That may be no bad thing.
Adil Rami, the strapping Valencia defender who was at the heart of the French back line alongside Mexes, was not much better as he continually and needlessly surrendered possession during key sections of the evening.
With Yohan Cabaye training separately from the rest of the French squad after collecting an injury sustained in the 2-0 win over Ukraine, Yann M'Vila came into the midfield for his first start of the finals.
Sweden coach Erik Hamren made two changes from the team who lost 3-2 against England knowing his team were already eliminated. Midfielder Emir Bajrami replaced the ill Rasmus Elm with Toivonen starting for the injured Johan Elmander up front.
In a first half of few chances, Toivonen could have handed Sweden a lead on four minutes when he rose to head wide of goal from Martin Olsson's floated cross.
The PSV Eindhoven forward should have hit the net on 12 minutes when he rolled Mexes before eluding France goalkeeper Hugo Lloris.
He seemed likely to score, but just overran the ball as he slipped a shot against the outside of the post with Lloris grounded.
Sweden could have added a second moments after Ibrahimovic scored with the productive substitute Christian Wilhemlsson and Olof Mellberg forcing to Lloris to make eye-catching saves to prevent France from falling out of the game.
Samir Nasri walloped a shot wide of goal on 63 minutes before limping off injured while M'Vila forced Andreas Isaksson to repel his shot 20 minutes from time, but France's best chance of the second half fell to substitute Olivier Giroud, who somehow headed wide from eight yards out on 84 minutes from Florent Malouda's corner.
The life seemed to have drained from France. That was confirmed when Larsson finished off the match in added time.
For the record, Karim Benzema has now attempted 15 shots without scoring, more than any other player at the tournament.