Captain Wes Morgan scored the only goal of the game with a header from 10-yards just before halftime.
The result moves Claudio Ranieri's men seven points clear of second placed Tottenham Hotspur which could only draw against Liverpool Saturday
With just six games of the season remaining, Leicester now require four more victories to guarantee a first top division title in its 132-year-history.
"We really wanted it today," a delighted Morgan told Sky Sports after the match.
"We saw the (Tottenham) game yesterday was a draw and wanted to make the most of our opportunity," he added.
Leicester had chances to increase its advantage in the second half. Southampton's England international goalkeeper pulled off a remarkable save when stretching to keep out a cross that had deflected off defender Jose Fonte.
Full-back Danny Simpson was then denied by Forster from point blank range when it looked easier to score from Jamie Vardy's cutback.
However, Southampton boss Ronald Koeman was aggrieved that his side was not awarded penalties for what he believed were two clear handballs.
The first saw the ball strike Simpson's hand as he scrambled to clear when Saido Mane had rounded Leicester keeper, Kasper Schmeichel, midway through the first half.
Leicester center back, Robert Huth, then appeared to handle when a strike from Charlie Austin cannoned off him in the second period.
"For the third match in a row we don't get a 100% penalty," a frustrated Koeman told Sky Sports.
Ranieri, by comparison, was ecstatic that his team had shown the spirit to maintain its most unlikely of title charges.
"Clean sheet, three points, step by step. It's fantastic because we know very well Southampton came here with a lot of confidence," he told Sky Sports.
"I enjoy it a lot when we maintain clean sheets. We are dreaming, if we fight we can do something and we must continue. It's important to stay very calm and maintain our feet on the ground."