The striker scored a brace to ensure that Nigel Pearson's side earned their sixth victory in seven matches, and climb up to 15th in the table with two games remaining
Algeria winger Mahrez struck twice during the first half to thrill a raucous King Power Stadium and secure a sixth win in seven matches for Leicester.
City's victory helped to confirm relegation for Burnley, despite the latter's 1-0 win at third-bottom Hull City, who are three points behind 15th-placed Leicester with two matches to play.
For Southampton, another lacklustre performances meant a sixth loss in their past 11 games, although defeat for Tottenham at Stoke City means Ronald Koeman's team remain one point shy of automatic qualification for the UEFA Europa League
Paulo Gazzaniga made his first start of the season in goal as one of four changes by Koeman, while Nigel Pearson stuck with the same Leicester XI that comprehensively beat Newcastle United last weekend.
Robert Huth looped an early header over amid a confident start from Leicester that was rewarded in the seventh minute.
There appeared to be little danger when Mahrez collected a loose ball centrally, but he bustled past Southampton midfielders Harrison Reed and Steven Davies to arrow a low drive into the bottom right-hand corner from 20 yards.
The buoyant mood at the King Power Stadium was checked by home midfielder Matty James going to ground with what appeared to be a serious knee injury, although the hosts doubled their advantage with 19 minutes played.
Southampton were unable to recover from a poor clearance by Gazzaniga and Jamie Vardy hooked the ball across goal for Mahrez to prod home left footed.
Sadio Mane passed up a chance to reduce the deficit, heading over when Davies nodded back into the six-yard box, but they should have been further behind 10 minutes before half-time.
Mahrez again got the better of the recalled Reed to slide a pass through to Vardy, only for Gazzaniga to come out on top in the one-on-one.
In the 58th minute, Mahrez had the opportunity to complete his hat-trick, but dragged wide left footed after Victor Wanyama misjudged a headed clearance.
Leicester's new-found defensive solidity ensured Southampton enjoyed few sights of goal during a more even second half, although Kasper Schmeichel was at full-stretch to keep out a dipping 20-yard effort from Mane inside the final 20 minutes.
Veteran midfielder Esteban Cambiasso almost crowned another influential display for Leicester by clipping a shot against the side netting, as Pearson's men closed out three points that put them on the brink of a relegation escape that seemed improbable a month ago.