Van Gaal’s men were already on the ropes and when Chris Smalling slipped before missing a header, Sturridge should have floored them but the England striker could only fire straight at De Gea.
United had apparently given the 2,000 or so travelling fans away shirts to create a white wall at the Anfield Road end.
Not many seemed bothered to wear them and it was more of a white flag waved by the players in the first half as Adam Lallana then made it a hat-trick of good saves for De Gea with a volley.
The best United could muster before the break was a wild shot from Marcos Rojo which was met with much delight from a delirious Kop.
Morgan Schneiderlin got United’s first shot on target after the restart, but it barely troubled Reds keeper Simon Mignolet.
De Gea pulled off another spectacular stop soon after to tip Coutinho’s crack from distance over before he punched away a stinging strike from Clyne.
Jordan Henderson then turned one over after a Lallana cut back before the second finally arrived.
Michael Carrick made a mess of a cross from Henderson before Lallana crossed for Firmino to fire in.
Marouanne Fellaini was lucky to escape a red card at the death after felling Emre Can with an elbow which was not spotted by the referee.
Klopp will warn his side that 2-0 is a dangerous lead going to the second leg.
But they are well up on the scorecards in this bout and Van Gaal will have to pick his side up off the canvas to rally at Old Trafford.