The Saints, who had also gone nine matches without a home win prior to today, are now within touching distance of Swansea City ahead of the Swans' game in hand at home to Chelsea in Saturday's late kickoff.
Following a slow start to proceedings, the hosts created a couple of good openings in quick succession, neither of which they could take.
Southampton were rewarded with an opening goal 25 minutes in when Tadic guided the ball into the bottom corner, with Mario Lemina picking out his teammate following a swift counter.
Having been on top throughout the first half, and had a penalty appeal turned down when Austin went down following a collision with Begovic, Southampton could easily have been pegged back when Nathan Ake's poked shot was helped away by Alex McCarthy.
McCarthy kept out Cook in the first of the second-half chances, shortly before Tadic doubled his personal tally for the afternoon by stealing the ball off Cook and firing into the bottom corner with the outside of his boot.
With mistakes beginning to creep into their game late on, the home side were grateful to goalkeeper McCarthy for earning them all three points.
SOUTHAMPTON (4-4-2): McCarthy; Cedric, Yoshida, Hoedt, Bertrand; Tadic (Ward-Prowse 87'), Romeu, Lemina, Bednarek; Redmond (Long 59'), Austin (Hojbjerg 72')
BOURNEMOUTH (4-4-2): Begovic; Francis, S.Cook, Ake, Daniels; Mousset (Ibe 64'), Gosling (Surman 72'), L.Cook, Fraser; Wilson (Defoe 64'), King