Nathan Redmond missed a huge chance to make the breakthrough for Southampton in the fifth minute and the visitors were made to pay one minute later when Arsenal took the lead.
Indeed, Lacazette, who led the line with Pierre-Emerick Aubameyang on the bench, was on hand to turn a Mkhitaryan cross into the back of the net at the far post.
Once again the visitors were made to pay for the missed opportunity, with Mkhitaryan turning a low Alex Iwobi cross into the bottom corner to double Arsenal's advantage with 17 minutes on the clock.
A poor clearance from Saints goalkeeper Angus Gunn led to Iwobi having the chance to deliver, and it proved to be a disastrous start to the match for the struggling visitors.
Southampton were bright in the final third, particularly down the right with Yann Valery, but Arsenal were being afforded too much space in the wide areas, which continued to bring opportunities.
There were plenty of occasions when the Saints had the chance to deliver from wide areas, meanwhile, but without a recognised centre-forward, it was comfortable for Arsenal to defend their own penalty box.
Lacazette had an incredible chance to make it 3-0 Arsenal in the latter stages of the first period when the ball was presented to the striker inside the box, but the Frenchman somehow missed the crossbar from close range, before Gunn denied him with a smart save moments later.
Southampton boss Ralph Hasenhuttl introduced both Michael Obafemi and Charlie Austin at the interval, with Redmond moving into a wide area as the visitors desperately looked for a route back into the match.
James Ward-Prowse almost found the corner with a powerful strike in the 52nd minute during a good spell for Southampton, but the midfielder's effort just flashed wide of the post.
The Saints were the team playing the better football as the hour mark came and went, which led to Arsenal making their second change, with Mesut Ozil introduced in place of Aaron Ramsey, who is joining Juventus this summer.
Obafemi was forced off with a hamstring problem in the 65th minute, though, with Hasenhuttl forced to substitute the player that he only introduced at the start of the second period.
Pierre-Emile Hojbjerg had a shot deflected wide of the Arsenal post 20 minutes from time with Southampton looking to make it a nervy finish, but the damage was ultimately done in the first period.
Arsenal came extremely close to scoring a third in the latter stages when Aubameyang's low cross was deflected towards his own goal by Jannik Vestergaard, but Gunn made a reaction save, before Leno kept out a powerful Targett effort down the other end as it finished 2-0 to the home side.
Next up for Arsenal is a home game against Bournemouth on Wednesday, while Southampton will attempt to return to winning ways when they host fellow strugglers Fulham in what is a huge match on the same night.
ARSENAL (4-2-3-1): Leno; Lichtsteiner (Koscielny 56'), Sokratis, Mustafi, Kolasinac; Torreira, Xhaka; Mkhitaryan, Ramsey (Ozil 63'), Iwobi (Aubameyang 75'); Lacazette
SOUTHAMPTON (5-3-1-1): Gunn; Valery, Bednarek, Vestergaard, Stephens (Austin 46'), Targett; Hojbjerg, Romeu, Ward-Prowse; Armstrong (Obafemi 46', Elyounoussi 65')); Redmond