Elche captain Gonzalo Verdu was sent off after 14 minutes before Lewandowski and Depay put Xavi 2-0 up at half-time.
Lewandowski scored his second of the game after the break, making it 11 goals in all competitions since his arrival from Bayern Munich as Barca won comfortably.
The win continued Barca’s unbeaten start to the domestic season as they moved into top spot ahead of Monday’s (AEST) Madrid derby.
Verdu was sent off for cynically hauling down Lewandowski after Frankie de Jong had chipped the striker in the goal.
Edgar Badia made a number of saves to keep out the hosts, but the Catalan side eventually capitalized on their numerical advantage in the 34th minute when Lewandowski tapped in Alejandro Balde’s cross.
Depay made it 2-0 soon after, flying into the roof of the net after confounding John Donald with a brilliant turn.
Barca thought they were three ahead at the break when Pedri came on, but the midfielder was caught offside in the build-up.
Lewandowski scored the second goal just after the break, firing into the bottom left corner after the ball was played to him in the box.
The Poland international nearly completed his hat-trick, seeing the ball saved by Badia with his feet, before another effort flew over the crossbar.
But those misses didn’t matter much as Barca went on to claim another victory.