It was only Millwall's fourth home win of the season, but it was enough to give some cheer to the suffering home fans and give them some hope for next season's campaign in League One.
Burnley, however, looked to have one eye on the summer, as they only managed to muster two shots on target in the whole game.
Williams' quick instincts allowed him to grab the only goal of the game after latching onto a Burnley error.
Midfielder James O'Connor tried a back header to his keeper Brian Jensen, only for Williams to read and race through and lob over the Clarets keeper from the edge of the area.
Caretaker boss Alan McLeary chose to give 19-year-old Will Hendry his full debut and he almost made an immediate impact four minutes in.
Barry Cogan played a neat one-two with Berry Powel before crossing for Hendry who miscued his shot from ten yards out.
Cogan was again the provider on 19 minutes when his in swinging free-kick from the left found Marvin Elliott at the far post only for his tame header to be comfortably saved by Jensen.
Ben May nearly created an opening on the stroke of half-time after Tony Craig's hopeful long ball.
May held off his man and squared for Powel who was well challenged by Jon Harley with the goal gaping.
But seconds later the home side got their reward for persistence after an error by Burnley midfielder James O'Connor.
His misplaced header back towards his own area played in Williams and he made no mistake with a lob over the oncoming Jensen.
Millwall continued to dominate after the interval against a Burnley team who seemed to have their mind on the end of the season already.
Rarely threatening even in Millwall's half they were again on the back foot in the 64th minute from Tony Craig's corner.
Elliott headed back across the goal and May was unlucky to see his header hit the roof of the net from close range.
However, with just a minute remaining Burnley almost snatched unlikely point.
Alan Mahon's inswinging free-kick from the left was met by Frank Sinclair six yards out, only for his header to go straight at the Lions keeper Colin Doyle.