Redmond stars as Southampton reach FA Cup semi-finals
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Ralph Hasenhuttl's side have slumped into the Premier League relegation battle after losing 10 of their last 12 top-flight matches.
While Premier League survival is the priority, Hasenhuttl insists the FA Cup was a "big target" and he made just two changes against a Bournemouth team who sat seventh in the Championship.
It proved a wise move as Southampton carved open the Bournemouth defence to take the lead in the 37th minute.
Redmond's enterprising run took him past two defenders to the edge of the area and he slipped a deft pass to Djenepo, who guided a composed finish into the far corner from eight yards.
Having provided the assist for the Mali winger's first goal since October, Redmond did it all on his own to double the lead in first half stoppage-time.
Seizing possession after Diego Rico failed to deal with Stuart Armstrong's chested flick, Redmond accelerated towards the Saints area, leaving Steve Cook in his wake before guiding a superb strike into the roof of the net.
Redmond put the result beyond doubt with his third goal of the season in the 59th minute, the midfielder slotting home from 10 yards after Armstrong's shot bounced back off the post.
In the day's second quarter-final, Premier League leaders Manchester City head to Goodison Park as they aim to remain on course for an unprecedented clean-sweep of all four major trophies.
Pep Guardiola's side are 14 points clear at the top of the Premier League, face Tottenham in the League Cup final and have a Champions League quarter-final date with Borussia Dortmund.
In Sunday's last eight action, Manchester United travel to Leicester in a clash between the teams currently in second and third place in the Premier League respectively.
United are hoping to win the first trophy of Ole Gunnar Solskjaer's reign this season after four years without major silverware.
United last won the FA Cup in 2016, while Leicester have not reached the final since 1969 and are still waiting to win the competition for the first time.
Chelsea, beaten in last year's final by Arsenal, will look to extend Thomas Tuchel's club record unbeaten start as manager to 14 games when they host struggling Sheffield United.