Fermin Lopez headed Barcelona into a 22nd-minute lead, before Hugo Duro capitalised on a Marc-Andre ter Stegen mistake to restore parity.
Pepelu then kept his composure from the penalty spot to turn the game in Valencia's favour, but they were dealt a major setback when goalkeeper Giorgi Mamardashvili was sent off in first-half stoppage.
Lewandowski made full use of the numerical advantage, scoring two headers before curling home a free-kick to wrap up his hat-trick and Barcelona's fourth consecutive home win.
The win moves Blaugrana back into second position in La Liga, while Valencia remain in eighth place and one point adrift of the top seven.
Ter Stegen, Araujo make costly mistakes
Barcelona were to respond to the recent Clasico defeat in their first outing since Xavi confirmed he had made a U-turn and would stay remain at the club for next season.
Xavi saw his side exposed by Valencia's threat on the counter in the opening exchanges at Estadi Olímpic Lluís Companys.
Duro's flicked effort forced Ter Stegen into action, before Peter Gonzalez dragged the ball harmlessly wide from a promising one-on-one opportunity.
Valencia were punished for missing those two chances when Fermin rose high in the box to power a header home from Raphinha's inviting cross.
However, Barcelona led for just four minutes before Ter Stegen carelessly lost the ball to Duro on the edge of the box, allowing the Valencia striker to roll home his 13th league goal of the season.
Ronald Araujo then gifted Los Che the chance to go to the penalty spot after he was penalised for a rash challenge on Peter.
Pepelu shouldered the responsibility to dispatch the subsequent spot-kick, leaving Barcelona with significant work to do.
Mamardashvili sees red
Joao Cancelo attempted to lead the Barcelona response, drawing a superb near-post save from Mamardashvili in the Valencia goal.
Just moments after making a fine stop, Mamardashvili made a costly error which changed the complexion of the contest.
The Georgian goalkeeper was given his marching orders for handling the ball outside of his box, preventing Lamine Yamal from running through to an unguarded net.
Andre Almeida was sacrificed so Jaume Domenech could take his place in goal and make a strong stop from Raphinha's free-kick, ensuring Los Che took a narrow lead into the break.
Lewandowski leads Barcelona fightback
Unfortunately for Ruben Baraja's youthful side, their slender advantage was wiped out within four minutes of the restart.
Lewandowski towered high to meet Ilkay Gundogan's near-post corner and send his glancing header beyond Domenech.
As the contest moved into the final 20 minutes, Barcelona threatened to score from another corner, but Domenech was ultimately equal to Araujo's header.
Diego Lopez raced up the other end of the pitch to bear down on Ter Stegen's goal, before his opportunity to unleash a strike was taken away by Pedri's last-ditch challenge.
Lewandowski proved to be Barcelona's hero, nodding in from close range after Domenech failed to deal with a corner.
The Poland international made sure of the three points in the third minute of stoppage time, curling a free-kick over the wall and out of the reach of Domenech to complete his first-ever La Liga hat-trick.
While the title may be out of reach, the win keeps Barcelona on track to secure second place, with two points separating them from Girona.
The Blaugrana will face Michel's side in a crucial game on Saturday, while Valencia will have to pick themselves up for Sunday's home clash with Alaves.