Lacazatte opened goal in the first half, reacting quickly to head a rebound into the net after Alexis Sanchez hit the post with a freekick.
He completed his brace from the penalty spot after Aaron Ramsey was fouled in the West Brom’s 18-yard-box.
After the game, the France international was selected for a random drug test which he took to Twitter to joke about.
“2 goals so.. doping control ?? well done guys,” the 26-year-old wrote on Twitter.
His manager Arsene Wenger was very pleased with the striker’s performance.
“I believe that he is getting stronger in every game,” Wenger said after the game.
“He is young and just arrived in England. The physical demands are very high.
“Technically, he is intelligent and you can see on the free-kick, he was there for the rebound like all good strikers are.”
The win takes Arsenal to seventh in the Premier League.