With leading scorer Lionel Messi suspended for the first time in his career, the champions lacked a cutting edge until Andres Iniesta fired them in front just before halftime at the Nou Camp.
The game was turned on its head a minute into the second half when home defender Gerard Pique was shown a harsh red card as he got in a tangle with Miguel de las Cuevas on the edge of the Barca area.
Iniesta then conjured a third goal for Xavi who finished with a neat lob at the end as Barca closed the gap on leaders Real Madrid to seven points before Jose Mourinho's side hosted Espanyol at the Bernabeu on Sunday.
"With Messi it would have been easier because he is the best player in the world," Mali international Keita told Spanish television.
"I don't know if I have saved the team. It was just important to win three points."
"I have seen it on television and I didn't even commit a foul," he said. "It felt like it was a premeditated decision.
"I told the referee at half-time he should have signalled a penalty in favour of Keita and he remembered this."
Manuel Pellegrini's side climbed into the Champions League qualification places with a 3-1 win at Getafe courtesy of three spectacular long-range strikes.
Portugal's Eliseu fired Malaga level after 57 minutes, France midfielder Jeremy Toulalan crashed a second in off the post and Spain's Santi Cazorla caught out the home goalkeeper from just over the halfway line with a wonderful lob in stoppage-time.
Atletico Madrid stayed ninth with 33 points after being held to a 1-1 draw at tenth-placed Sevilla, a point doing little for either side's European aspirations.
