Barcelona players celebrate their win (Photo: AP)
Champions Real Madrid's win at Osasuna the day before pulled them closer to Xavi Hernandez's side, who had one eye on Thursday's Europa League play-off second leg against Manchester United at Old Trafford.
However an inspired Ferran Torres helped Sergi Roberto to break the deadlock and Lewandowski added a second to win the game and leave Cadiz, 17th, two points above the relegation zone.
The visitors had two goals disallowed and made Barcelona uncomfortable at times at Camp Nou but their victory rarely seemed in doubt.
Xavi began with Sergi Roberto in midfield, replacing Pedri, who suffered a thigh injury in the Europa League 2-2 first leg draw with Manchester United last Thursday.
However it was Torres' arrival that made the difference for Barcelona in the first half.
The winger has struggled badly at times this season, largely left out of the picture, but with Ousmane Dembele injured and Raphinha rested to face United, the former Manchester City forward took his chance with both hands.
Torres was hyperactive, dribbling past defenders at pace and tying Cadiz's defence in knots.
One such run down the right saw him cross for Lewandowski after 43 minutes and although the Polish striker's effort was saved, Sergi Roberto was on hand to smash home the rebound.
Three minutes later Barcelona doubled their lead when Lewandowski collected the ball on the edge of the Cadiz box and fired a low strike into the bottom corner past Jeremias Ledesma.
Barcelona took their foot off the gas in the second half, although Lewandowski came close to a second with a lobbed effort that clipped the top of the crossbar.
La Liga's top scorer with 15 goals, Lewandowski has not been at his sharpest since the World Cup but slowly seems to be returning to it.
Cadiz striker Roger Marti, who had a goal ruled out for offside early on, was infuriated as another was struck from the record in the second half.
Barcelona goalkeeper Marc-Andre ter Stegen was challenged by Cadiz defender Fali before Roger shot and the referee deemed the contact enough to be a foul.
Ter Stegen made a fine save to deny substitute Chris Ramos at the near post and keep yet another clean sheet, with Barcelona having conceded just seven goals from 22 matches in La Liga.
In Europe it has been another story, with Xavi hoping his side can prove more sturdy at the back against Man United than they were in the first leg.
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