At a boisterous Anfield, Liverpool were on top from beginning to end as United lost to their great rivals for the first time under the stewardship of Louis van Gaal, a penalty from Daniel Sturridge and a Roberto Firmino effort separating the teams.
Jurgen Klopp’s side will go to Old Trafford for the second leg next Thursday in a good position to advance to the quarter-finals, although that United retain any hope is largely down to David de Gea.
The Spanish goalkeeper could not keep out Sturridge’s 20th-minute penalty that followed a tangle between Memphis Depay and Nathaniel Clyne just inside the box, but he kept the visitors in the game for long periods after that.
De Gea saved twice from Philippe Coutinho and once each from Sturridge and Clyne before Liverpool, attacking the Kop with gusto in the second half, got a deserved second goal in the 73rd minute.
When Michael Carrick, who had replaced the suffering Marcus Rashford at half-time, made a mess of dealing with a Jordan Henderson ball, the excellent Adam Lallana pounced to set up Firmino to score.
“We can be confident with a two-goal advantage but that can be dangerous sometimes,” Henderson told BT Sport as he looked ahead to the return.
“We’ve just got to go there as if it were 0-0 and take it from there.”
Earlier, a brilliant Dortmund brushed aside a Tottenham side not helped by coach Mauricio Pochettino’s decision to rotate his squad.
For his 100th game in charge, Pochettino made seven changes to the team that drew 2-2 with Arsenal last weekend with Harry Kane among those left on the bench and Dele Alli suspended.
Teenager Joshua Onomah was among those handed a start against a Dortmund side near full strength, although Ilkay Gundogan was missing with a foot injury.
The prolific Pierre-Emerick Aubameyang had one effort ruled out for offside before getting the breakthrough on the half-hour mark, rising above Ben Davies to head home a Marcel Schmelzer cross.
Tottenham were never at the races, and Gonzalo Castro hit the post from a tight angle in the 51st minute after Aubameyang had lifted the ball over Hugo Lloris.
– Reus double –
Dortmund struck again on the hour mark, Neven Subotic heading on a Castro cross for Germany star Marco Reus to lash home on the volley at the back post.
And they appeared to have the tie sewn up as Reus scored his second of the evening with 20 minutes remaining, stroking past Lloris from close range after superb build-up play involving Aubameyang and Castro.
“It is difficult to explain but it’s a bad day and I’m disappointed with our performance. We did not play well. That is why we lost the game,” said Pochettino.
Elsewhere, holders Sevilla drew 0-0 away to Basel in Switzerland and ended the game with 10 men following the late sending-off of midfielder Steven N’Zonzi.
In an all-Spanish tie, Athletic Bilbao edged Gary Neville’s Valencia 1-0 on a sodden San Mames pitch with Raul Garcia scoring the only goal of the game.
Villarreal are the best placed of all Spain’s representatives, though, after a 2-0 win at home to Bayer Leverkusen at El Madrigal secured thanks to a goal in each half by Cedric Bakambu.
Leverkusen, who had Tin Jedvaj sent off late on, now have it all to do in the second leg in Germany.
Shakhtar Donetsk, UEFA Cup winners in 2009, are in a strong position after beating Anderlecht 3-1 in Ukraine.
Taison and Olexandr Kucher struck in the first period for Shakhtar and Eduardo added a third late on after Frank Acheampong pulled one back for the Belgians.
Lazio came from behind to draw 1-1 with Sparta Prague in the Czech Republic as a Martin Frydek opener was cancelled out by Marco Parolo.
And Fenerbahce have the edge against Portuguese side Braga following a 1-0 win in Istanbul given to them by a late Mehmet Topal goal.