Manager Mark Hughes recorded his first league victory as Southampton manager with his side showing fighting spirit at just the right time, moving to 32 points and within a point of the safety zone with three games left.
With mistakes beginning to creep into their game late on, the home side were grateful to goalkeeper McCarthy for earning them all three points.
AFC Bournemouth went into half-time level when Joshua King converted a corner at the far post.
"It's been a challenge for everybody to get back to winning ways, it's been a long time, too long given the quality we have".
Defensive errors had become the theme of the match and in the 54th minute it was Cook's turn to be the charitable party, the 27-year-old's poor chest control on the halfway line allowing Tadic to stride in on goal and stroke a classy finish into the bottom left corner.
The win preserves hope, and a trip to Everton is next on the schedule.
Dusan Tadic has urged the Southampton board to retain Mark Hughes as manager beyond the end of the season after the club improved their survival chances by beating Bournemouth.
- The Cherries have now conceded at least 60 goals in all three of their seasons in the Premier League. Tadic, who scored twice in Southampton's first home win since November, has credited Hughes with galvanising the club since his appointment on a short-term contract in March.
Southampton have never lost at home to Bournemouth in any competition (W8, D5).