Bournemouth's home defeat by Southampton means they now must win at Everton on the final day of the season to have any chance of avoiding Premier League relegation.
Substitute Sam Surridge thought he had snatched a point deep in added time only for his goal to be correctly ruled out by the video assistant referee because Callum Wilson was offside in the build-up.
Che Adams then scored in the ninth minute of injury time to add to Danny Ings' first-half strike, who also missed a penalty, and give the scoreline an unfair slant.
Even a Cherries win in their final game might not be enough, as they also need Watford to lose to both Manchester City and Arsenal, and for their to be a big enough goal difference swing.
Wilson and David Brooks were recalled as relegation-threatened Bournemouth made two changes, with Dan Gosling and Dominic Solanke dropping to the Cherries' bench following Wednesday's narrow defeat at Manchester City.
Saints boss Ralph Hasenhuttl opted for five alterations to the team which started Thursday's 1-1 draw with Brighton.
Jack Stephens, Ryan Bertrand, Kyle Walker-Peters, Stuart Armstrong and Shane Long came in, with Pierre-Emile Hojbjerg among those making way.
Ings' 21st top-flight goal of the season, from the edge of the box, put Southampton ahead four minutes before half-time and he squandered an excellent chance to double Saints' advantage in the 59th minute.
After VAR penalised Cherries substitute Harry Wilson for handball, Ings' tame penalty was saved by Aaron Ramsdale.
Ramsdale made another excellent save to deny Nathan Redmond in the closing stages but there was no breakthrough to be found at the other end as Solanke had a shot blocked by Jannik Vestergaard and Alex McCarthy saved well from Harry Wilson.
Bournemouth thought they had a point only for Surridge to be denied, and there was still time for Adams to fire home for Saints in the final seconds to put the result beyond doubt.
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