Messi scored twice in the first half to lead Barca to a comprehensive 3-0 second-leg victory at Camp Nou and through to the semi-finals of the competition.
The five-time Ballon d'Or winner was kept quiet in Barca's 1-0 win in Manchester last week but he was lethal on Tuesday – a pair of goals in the space of three minutes extinguishing United's hopes.
Philippe Coutinho capped a 4-0 aggregate triumph with a stunning long-range effort on the hour and United attacker Lingard said to BT Sport: "You can't afford to make mistakes in these games.
"We dominated the first 15 or 20 minutes, and it could have been different if we had put our chances away.
"Messi is one of the best players in the world and he brought his A game tonight. All we can do now is move on.
"We were always going to be underdogs, but we had to back ourselves and believe. If the first 15 minutes had gone our way, it would have been a different game.
"Chances in these games are massive and if we take them it's a different game. The Champions League is a big competition and we'll be fighting to get back in."