Austin’s goals came six minutes apart with the first restoring Southampton’s lead, which they had lost on the stroke of halftime.
Dusan Tadic also found the net for his first goal of the season whilst Steven Davis made it a comprehensive and vital victory.
Southampton entered having scored just nine goals this season, but found the target four times to move into 10th place in the table.
Austin nearly opened the scoring inside a quarter hour only to see his effort catch the outside of the post on 12 minutes.
But Tadic gave the hosts a deserved lead on 18 minutes when he collected the ball out of his feet before slipping it past Jordan Pickford.
Southampton continued to dominate proceedings, but Gylfi Sigurdsson’s netted his first goal for Everton to level the match on the stroke of half-time.
Saints, however, regained their composure and Austin stepped up to restore the lead.
The striker, who has been reduced to cameo appearances, met Ryan Bertrand’s cross with a fine header to make it 2-1.
Austin wasn’t done as he again finished with a header -- this time from Tadic -- to double the advantage.
Davis would make it 4-1 nearing full-time as he found the back of the net from the edge of the box.
Southampton: Forster, Bertrand, Hoedt, van Dijk, Cedric, Hojbjerg, Davis, Ward-Prowse, Boufal, Tadic (Lemina 78), Austin (Long 80).
Unused subs: McCarthy, Yoshida, McQueen, Redmond, Gabbiadini.
Goals: Tadic (18), Austin (52, 56), Davis (87)
Everton: Pickford, Kenny, Baines (Williams 27), Keane (Vlasic 75), Jagielka, Schneiderlin, Gana Gueye, Mirallas (Lookman 64), Lennon, Sigurdsson, Calvert-Lewin.
Unused subs: Robles, Ramirez, Rooney, Baningime.
Goals: Sigurdsson (45).