United were reeling at halftime and Southampton’s James Ward-Prowse was set to hog the headlines after setting up Jan Bednarek’s opener and then scoring another sublime free kick.
But Uruguayan Cavani showed why United signed him from Paris St Germain in October, coming on to replace Mason Greenwood for the second half to devastating effect.
He picked out Bruno Fernandes on the hour to get United back in the game, then headed his side level in the 74th minute, and finally stooped to nod the winner in in the 92nd minute, sealing an eighth successive league win for his side.
United became the first side in Premier League history to come from behind to win four successive away games and they had Cavani to thank for the dramatic turnaround on the south coast.