Dortmund hammered in-form Tottenham Hotspur 3-0 in Germany, with Marco Rues netting twice in 10 minutes after Pierre-Emerick Aubameyang had headed in the opener on the half-hour-mark.
At Anfield, Liverpool beat Manchester United for the first time in five games, with Daniel Sturridge’s first half penalty and Roberto Firmino’s second half strike giving the Reds the bragging rights.
In fact, United have goalkeeper David de Gea to thank for still being in the tie, with the Spaniard pulling off a succession of fine saves to keep the hosts at bay.
Gary Neville’s Valencia lost again in an all Spanish tie, with Athletic Bilbao's Raul Garcia getting the winner on 20 minutes.
Villarreal also won, with Cedric Bakambu’s brace doing the business against Bayer Leverkusen, who had Tim Jedvaj sent off in stoppage time.
Defending champions Sevilla were held to a goalless draw at Basel, while Marco Parolo ensured Lazio escaped with a 1-1 draw at Sparta Prague.
Shaktar Donetsk 3 Anderlecht 0
Basel 0 Sevilla 0
Borussia Dortmund 3 Tottenham Hotspur 0
Fenerbahce 1 Sporting Braga 0
Villarreal 2 Bayer Leverkusen 0
Athletic Bilbao 1 Valencia 0
Liverpool 2 Manchester United 0
Sparta Prague 1 Lazio 1