The game got off to a disappointing start for Barcelona, with Ferran Torres forced to make in the fifth minute due to an injury issue.
However, the visitors' mood quickly improved when Lewandowski headed home from Raphinha's free-kick to register his 50th La Liga goal.
Lewandowski then linked up with Raphinha again to double Barcelona's advantage, before he drove a low shot into the bottom corner to complete the fourth-fastest La Liga hat-trick in Barcelona's history.
Alaves thought they had gained a precious foothold when Toni Martinez headed home from Nahuel Tenaglia's whipped cross, only to see the effort ruled out for offside.
The Basque side continued to offer a threat in the second half as Barcelona's high line encouraged them to push forward in search of a way back into the contest.
However, Barcelona's defensive line ultimately came out on top, especially as Alaves saw another goal chalked off for an offside infringement.
Hansi Flick's side had chances of their own in the second period, but they were content to settle for a comprehensive 3-0 victory, ensuring they will head into the international break with a three-point lead at the top of the table.
As for Alaves, they will finish matchday nine in 12th position after falling to a third consecutive defeat and a second straight loss without scoring.
ALAVES VS. BARCELONA HIGHLIGHTS
Lewandowski goal vs. Alaves (7th min, Alaves 0-1 Barcelona)
Lewandowski finds the early breakthrough at Mendizorroza. The Poland international produces a glancing header to steer the ball home from Raphinha's inswinging corner.
Lewandowski goal vs. Alaves (22nd min, Alaves 0-2 Barcelona)
Raphinha and Lewandowski combine again to double Barcelona's advantage. The Brazilian leads out a swift Barcelona counterattack, driving into the box before choosing the opportune moment to slide Lewandowski in for a simple finish.
Lewandowski goal vs. Alaves (32nd min, Alaves 0-3 Barcelona)
There's the hat trick for the in-form Lewandowski. After collecting Garcia's through ball, Lewandowski sends a low strike out of the reach of Antonio Sivera to complete an impressive hat-trick.
MAN OF THE MATCH - ROBERT LEWANDOWSKI
Lewandowski might be 36 years of age, but he appears to be in as good form as ever after producing another goalscoring clinic.
The Poland international scored three goals from his five shots on target, taking his tally for the season to 10 goals in nine La Liga matches.
As well as netting a lighting quick hat-trick, Lewandowski also won seven duels during the contest - more than any other player on the pitch at Mendizorroza.
ALAVES VS. BARCELONA MATCH STATS
Possession: Alaves 28%-72% Barcelona
Shots: Alaves 11-14 Barcelona
Shots on target: Alaves 2-9 Barcelona
Corners: Alaves 11-3 Barcelona
Fouls: Alaves 13-8 Barcelona
BEST STATS
WHAT NEXT?
Barcelona will head into the international break in high spirits, and they will be hoping to claim a third consecutive win in all competitions when they return to action against Sevilla on October 20.
As for Alaves, they will welcome newly-promoted Real Valladolid to Mendizorroza in the first game of matchday 10 on October 18.