The Spanish side led after eight minutes when Youssef En-Nesyri finished following an error from Harry Maguire.
Sevilla doubled their advantage with a looping header from defender Loic Bade, before En-Nesyri again capitalised on a mistake – from goalkeeper David de Gea – to complete the win.
United were well below their best as they exited Europe with a whimper.
This is the first of a series of potentially season-defining games for United – they take on Brighton at Wembley in the FA Cup semi-finals on Sunday and face a key Premier League match at Tottenham on 27 April – and Erik Ten Hag’s side completely failed this test.
Sevilla meanwhile will play Juventus in a two-legged semi-final on 11 and 18 May. The Italian side beat Sporting 2-1 on aggregate in their quarter-final.