Even before today the Clarets had made their best start to a top-flight season since 1973, and they maintained their remarkable form as Sam Vokes scored the only goal of the game with just nine minutes remaining.
The result leaves Burnley level on points with the likes of Chelsea and Arsenal - and three above Liverpool - whereas Southampton drop down to 13th having now won just three of their last 12 Premier League home games.
The opening exchanges were short of clear chances for either side, and a couple of Nathan Redmond efforts - the first of which was nodded tamely over and the second of which flew well over the crossbar - were the only attacking moments of note in the opening half-hour.
Southampton's threat was growing and Burnley needed a fine block from James Tarkowski in the latter stages of the half before Pope thwarted Redmond with the best save of the lot.
Burnley were relieved to hear the half-time whistle, but Southampton started the second half on the front foot too and Manolo Gabbiadini was the first to threaten with a header which bobbled wide.
Pope was proving to be the Burnley hero, though, and shortly afterwards he made another fine save to deny Maya Yoshida's goalbound effort from finding the top corner.
Sean Dyche turned to Vokes off the bench in an attempt to turn the tide, and it was a move that paid dividends nine minutes from time when he rose to meet a cross from Johann Berg Gudmundsson before planting a firm header past Fraser Forster from 12 yards.
Southampton had no response in the closing stages as Burnley held on for their first ever top-flight win away to the Saints, making it 11 points from their six outings on the road this season having only managed seven throughout the whole of last term. body check tags ::