Jan Bednarek's header and a trademark free-kick from James Ward-Prowse had put Southampton in command at St Mary's Stadium.
But the introduction of Cavani for the second half proved a masterstroke by Ole Gunnar Solskjaer that paid initial dividends in the 59th minute when he teed up Bruno Fernandes to halve the deficit.
He then met the midfielder's deflected shot with a diving header to equalise and had the crucial final say when he diverted Marcus Rashford's cross beyond Alex McCarthy in the 92nd minute, securing a record eighth successive Premier League away win for United.
United should have been ahead in the seventh minute when Alex Telles played Mason Greenwood through after poor defending from Jannik Vestergaard, however, the striker could only hit the side-netting after rounding McCarthy.
The visitors were in the ascendancy early on but went behind against the run of play as Bednarek glanced home an excellent corner to the near-post from Ward-Prowse.
Kyle Walker-Peters then saw an effort deflected onto the left-hand post before McCarthy atoned for his own error to prevent Fernandes from finding an equaliser.
Another pinpoint set-piece from Ward-Prowse punished United for their failure to take that chance to restore parity, his ninth goal from a direct free-kick in the Premier League squeaking inside the near-post despite David de Gea getting a hand to it.
De Gea then denied Moussa Djenepo after a quick break but received treatment having appeared to injure himself when he clattered into the post in a vain attempt to keep out Ward-Prowse's free-kick and was eventually replaced by Dean Henderson after half-time.
Cavani also entered the fray and had the desired impact as he picked out Fernandes to convert on the turn.
Fifteen minutes later Cavani levelled matters with a show of clinical finishing that he repeated in injury time as he stooped to head home Rashford's delivery and send United seventh in the table.
United's road warriors close on top four
United are the first team in Premier League history to win four consecutive away games despite falling behind in each. This most recent show of resolve moves them just two points behind fourth-placed Leicester City, who face Fulham on Monday, and a point adrift of Southampton in fifth.
