Club-record signing Lacazette netted a goal in either half - including one from the penalty spot - as the Gunners moved into seventh position in the Premier League table.
It was not all one-way traffic, however, with West Brom having a huge penalty appeal turned down in the early stages, before missing a number of excellent chances during an open first 45 minutes.
Arsenal's first sight of the West Brom goal came in the fifth minute when Sanchez stood over a free kick close to the away side's penalty area, and the Chilean's effort dropped just wide of Ben Foster's post.
West Brom should have had a penalty in the eighth minute when Shkodran Mustafi clearly brought Rodriguez to the ground with a rash challenge, but the referee seemingly played an advantage. Rodriguez's low effort was then turned onto the post by Arsenal goalkeeper Petr Cech, before Livermore fired wide from the rebound.
Arsenal then came close down the other end through Aaron Ramsey after the Welshman met a cross from Hector Bellerin, but the midfielder's effort just missed the crossbar during an entertaining spell at the Emirates.
The opening goal was always coming as the tempo continued to rise, and it was Arsenal that made the breakthrough in the 20th minute when Lacazette headed home from close range after Foster had only partially saved a Sanchez free kick.
Ramsey was next to come close after darting into the West Brom penalty area and rounding Foster, but the Baggies goalkeeper managed to save the Welshman's effort from a wide position. The Gunners then had a penalty shout of their own as Grzegorz Krychowiak appeared to tug Sanchez's shirt inside the box.
West Brom had a golden chance to level the scores in the 38th minute after Krychowiak broke towards the Arsenal box before delivering a super cross onto the head of Rodriguez, but the forward's effort was cleared off the line by Monreal as the Gunners escaped another dangerous moment.
Robson-Kanu was next to come close for West Brom as he headed wide of the post late in the first period, before Rodriguez missed another opportunity from close range as the Baggies somehow entered the interval behind.
Lacazette came close to sending Arsenal 2-0 ahead early in the second half after collecting a smart flick from Elneny, but the striker's effort was deflected just over the West Brom crossbar by Kieran Gibbs.
Arsenal continued to enjoy more of the ball at the Emirates, but the match headed for its final 30 minutes still at 1-0, which led to frustration and concern among the home supporters.
James Morrison and Salomon Rondon both came off the West Brom bench just past the hour, but there was no movement from Arsenal as the likes of Theo Walcott, Olivier Giroud and Ozil waited for their chance.
Arsenal had their second of the night in the 67th minute, however, when Lacazette found the bottom corner from the penalty spot after Nyom had bundled Ramsey to the ground inside the penalty area.
Lacazette had a decent chance to complete his hat-trick 20 minutes from time after collecting a brilliant pass from Sanchez inside the West Brom box, but the Frenchman missed the post from a wide angle.
Giroud and Ozil both came off the Arsenal bench in the 83rd minute, but aside from a Granit Xhaka effort which missed the crossbar, chances were few and far between late on as Lacazette's double proved the difference between the two teams on the night.
Arsenal will now switch their attention to the Europa League as they prepare to travel to BATE Borisov on Thursday night. Next up for West Brom, meanwhile, is a Premier League clash at home to Watford on Saturday afternoon.