Arsene Wenger’s side rode their luck in the first half and have Petr Cech to thank for keeping the score at 0-0.
A draw would not have been enough for the Gunners, who were seven points behind Liverpool at the start of play.
However, Alexis Sanchez struck on the hour with a moment of magic. After being slid in by Mesut Ozil, the Chilean’s fake shot sent two defenders for the St Mary’s exit, before he applied a powerful low shot passed Fraser Forster.
Substitute Olivier Giroud sealed the win late on after being teed up by Aaron Ramsey for a header.
Arsenal have climbed above Manchester United into 5th and are three-points behind fourth placed Manchester City. Though, third place Liverpool are the more likely target with the Reds having played a game more.