The Uruguay international came on at the break, with the hosts 2-0 ahead, thanks to Jan Bednarek and James Ward-Prowse, and set up Bruno Fernandes, for yet another goal.
The Portuguese midfielder then turned provider as his cross-cum-shot was headed home brilliantly by Cavani and the striker stooped again in stoppage time to earn a glorious victory.
FIRST HALF – SAINTS ON TOP
Ole made two changes to the side that swept past Istanbul Basaksehir in midweek as Mason Greenwood replaced Anthony Martial, who fell ill the night before the match, and Nemanja Matic came in for Cavani, who was named on the bench.