With a 1-0 advantage from the first meeting at Old Trafford, the Catalan side took command inside the first 20 minutes thanks to a double from its number 10, the second of which owed plenty to a dreadful mistake from David de Gea.
Ole Gunnar Solskjaer named an attacking line-up featuring Anthony Martial, Jesse Lingard and Marcus Rashford, but the blow of those first-half goals seemed to drain any hope of repeating the kind of performance it produced to stun Paris Saint-Germain in France in the last 16.
The tie was firmly put to bed by a sublime goal from Philippe Coutinho, who may get the chance to face old club Liverpool in the semi-finals if it sees off Porto on Thursday (NZST).
Rashford prodded a shot against the bar in a positive start from United, but they engineered their own downfall within the space of four minutes.
Ashley Young dithered in possession and lost the ball near his own box, allowing Messi to step into space and drill a shot beyond De Gea and into the bottom-left corner.
The United keeper could do little about it, but he endured a calamitous moment 20 minutes in, allowing Messi's tame right-foot shot to squeeze beneath his body and into the net.
De Gea denied a certain Sergi Roberto goal after Messi left Phil Jones chasing shadows, but he was again helpless as Barca struck a third on the hour mark, Coutinho sending a stunning strike beyond his outstretched right hand.
Marc-Andre ter Stegen made a magnificent 90th-minute save from substitute Alexis Sanchez, but United was well beaten and must now fight to turn around its recent form after a fifth defeat in seven games in all competitions.