It took just six minutes for Lacazette to convert from Mkhitaryan's attempted shot and the latter turned home the dangerous Alex Iwobi's assist at Emirates Stadium.
Lacazette wasted a couple of golden opportunities to stretch the lead before half-time and, while Arsenal's tempo noticeably dropped after the break, Unai Emery's side was in complete control.
Manchester United's goalless draw with rival Liverpool consequently sent Arsenal back into the UEFA Champions League places, while Southampton remains mired in relegation trouble.
Nathan Redmond failed to capitalise on lax Arsenal defending when shooting straight at Bernd Leno one-on-one and shortly afterwards the Gunners went in front.
Iwobi's left-wing cross travelled across the box, with Lucas Torreira seemingly felled by Jack Stephens in his bid to meet the ball, and Mkhitaryan's cross-cum-shot was cleverly flicked in by Lacazette.
It was 2-0 after 17 minutes as goalkeeper Angus Gunn was played into trouble by Stephens and Iwobi pounced on his hasty clearance before squaring for Mkhitaryan to side-foot home from the edge of the box.
Lacazette wastefully blasted over from six yards shortly before the break, while James Ward-Prowse fired wide on the volley for Southampton from Granit Xhaka's wayward pass early in the second half.
The chances began to dry up, although substitute Pierre-Emerick Aubameyang forced a great late save from Gunn, and the only slight concern for Emery was the sight of Iwobi struggling with a calf problem 15 minutes from the end.
With United drawing against Liverpool and Chelsea featuring in the Carabao Cup final, Arsenal stole a march in the race for fourth. Saints, meanwhile, remain 18th and one point shy of safety.