Lewandowski scored in the seventh, 22nd and 32nd minutes to take his league tally to 10 goals in nine matches.
The Polish superstar has a total of 12 goals in 11 games in all competitions, and has scored eight times in his last five matches.
He was coming off a brace in Barcelona's 5-0 rout of Young Boys in the Champions League on Tuesday.
"We played a very good match from the first minute," Lewandowski said.
"When you score three goals in the first half, it gives you the ability to be a lot more in control in the second."
The victory moved Barcelona three points back in front of Real Madrid ahead of the international break.
Madrid had drawn level on points with the Catalan club with a 2-0 win over third-place Villarreal on Saturday.
Barcelona were coming off their first league loss of the season — a 4-2 defeat against Osasuna, after previously having won their first seven matches in the competition.
Lewandowski scored with close-range strikes after a pair of assists by Raphinha and another from Eric Garcia.
It was the third loss in a row for Alaves, who had no attempts on goal in the first half.
Barcelona lost forward Ferran Torres with an apparent muscle injury five minutes into the match.
Girona ended a six-match winless streak in all competitions with a 2-1 victory at home against Athletic Bilbao, with Cristhian Stuani scoring the winner by converting a penalty kick nine minutes into second-half stoppage time.
Before that, the match had been marked by three penalty saves by Girona goalkeeper Paulo Gazzaniga — one in the first half and two in the second after a video review required the last penalty to be retaken.
It was Athletic's first loss in six matches in all competitions.
