Albion now just have the Europa League and the Premier League to concentrate on.
A glaring error by goalkeeper Jason Steele in the first minute saw Mario Lemina stab the ball home – after which Albion had a massive 18 shots at goal with only two on target.
It will be Wolves who have a tap in to a semi-final place as it is they who play Coventry City at home in the sixth round.
Steele also missed an absolute sitter in added time when he went up for a corner.
Valentin Barco made his debut for the Albion late in the second half and provided some ammunition for second-half substitute Danny Welbeck.
Jakub Moder missed an easy chance in the first half after industrious work by Simon Adingra.
Albion had almost all the play. Facundo Buonanotte was pulling the strings in midfield but he also missed an easy chance.
But the sickener for Albion fans will be that miss from Steele at the death.
Next up, Brighton travel to Craven Cottage to face Fulham in the Premier League on Saturday (2 March) when they can focus on qualifying for European competition again next season.
