Teenager Fati showed pace and anticipation to nip in between two Celta defenders and fire Barca ahead in the 11th minute, picking up where he left off with his two early goals in Sunday's 4-0 win over Villarreal.
Barca's lead looked to be under threat when Lenglet received a second booking at the end of the first half for striking former Barca player Denis Suarez in the face, sparking furious protests from Messi and Gerard Pique.
But Barca quickly doubled their advantage soon after the interval, as Messi tore his way through Celta's defence and fired a low shot which took a heavy deflection off Lucas Olaza and squirmed past keeper Ivan Villar for an own goal.
Messi almost got a goal for himself with another bursting run into the area but his shot was saved by Villar, leaving Sergi Roberto to lash high into the net deep in added time for the third goal. (Reporting by Richard Martin Editing by Toby Davis)