On a rainy Monday night in Catalonia, the home side dominated possession but were behind at the break due to two defensive errors.
Fermin Lopez put Barcelona ahead with a lovely glancing header midway through the first half.
Valencia levelled just five minutes later thanks to a woeful intervention from keeper Marc Ter Stegen. He intercepted a long ball 25 yards from his goal but he tried to be too fancy and Hugo Duro easily dispossessed him and ran the ball into the empty net.
Before the break, Valencia took the lead thanks to another elementary mistake. Ronald Araujo's mistimed tackle felled Peter Federico in the box and Pepelu coolly converted the spot kick.
The balance of the match changed moments before halftime when Mamardashvili used a hand outside the box to thwart a scoring chance by Lamine Yamal and was red-carded.
Lewandoski headed an equaliser from a corner kick after 50 minutes. However, it took Barca until the 82nd to regain the lead from another Lewandowski header after Valencia failed to clear another corner.
The hat-trick came in the third minute of added time from a 25-yard free kick.
"It wasn't easy," Lewandowski said.
"In the first half we didn't find any space in attack."
Barcelona replaced Girona in second place and are 11 points behind Real Madrid. Madrid need five points from their last five matches to guarantee a 36th Spanish league title.
Valencia remain eighth, four points adrift of the final European place.
