The Gunners had started brightly but took the lead through controversial fashion, after Santi Cazorla flung himself to the floor following a challenge by Steven Reid when replays showed there was no contact.
The Baggies, who lost twice last week to halt their fine start to the season, never really recovered from the sense of injustice and there was no way back once Arteta converted from the penalty spot again on 64 minutes after Chris Brunt had tripped Alex Oxlade-Chamberlain.
It was a lively start by the home side, who had not won in four games, but recalled most of their first-team regulars missing from the midweek Champions League defeat at Olympiacos, although forward Theo Walcott was sidelined by a calf injury.
A high pass into the West Brom penalty area was palmed away by Boaz Myhill, but Gervinho missed his kick and could only hack the ball wide from six yards. West Brom were forced into an early change when defender Liam Ridgewell hobbled off, replaced by Goran Popov.
Arsenal were awarded a penalty on 24 minutes. Cazorla cut into the left side of the area, and as he turned back inside Reid, went to ground as the defender stuck out his leg. Referee Mike Jones immediately pointed to the spot, although television replays showed there looked little contact, if any at all. Arteta stepped up to drill the ball past Myhill.
In the second half, the Gunners went forwards quickly on the counter, with Oxlade-Chamberlain floating a ball across the face of goal before Olivier Giroud and Olsson squared up on the edge of the penalty area, with the referee soon restoring calm after showing both players a yellow card.
On 64 minutes, Arsenal were awarded another penalty. This time there was no controversy as Brunt chopped down Oxlade-Chamberlain after the England midfielder had charged down the right. Arteta took it again and with the same outcome as the Spaniard blasted the spot-kick down the middle.
With a two-goal cushion, Arsenal had some breathing space and started to produce slick, passing football. Lukas Podolski came off the bench to somehow lift the ball over from six yards as Arsenal closed out what was in the end a comfortable victory.