The France global has not played since suffering the problem in a 2-2 home draw against Celta Vigo on December 2.
On 13 minutes, the No. 10 had Barcelona in front after he converted a cutback from Alba on the half-volley.
It was a vintage Messi display, too, and despite netting a couple of goals early on, the finest moment of the first half was his involvement in the third.
Suarez, who has found his groove after a boring start to season, shot the ball high into the net in the 38th minute off a cross from Sergi Roberto to score his fifth goal in four games. And once again, it was that linkup with Alba that Celta were unable to stem.
Luis Suarez took advantage of a poor back pass three minutes later to make it 4-0 and end any chance of Celta getting back into the tie.
The tie petered out in the second half but French worldwide Ousmane Dembele showed why Barcelona paid 105 million euros for him a year ago when he dribbled past three defenders from the halfway line before being denied by a desperate tackle.
After Messi came off to applause before the hour mark, Ivan Rakitic wrapped up the result late on with a free header following a corner.
Samuel Umtiti returned to training this week but remains unavailable with a hamstring injury, whilst reported £142 million signing Philippe Coutinho is out for around three weeks with a thigh problem.
And if Messi continues to perform to such high standards, it's hard to see any side halting the Blaugrana in any competition.
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