Redmond also set up Moussa Djenepo for the opening goal, which gave Saints a deserved lead on 37 minutes after having an early tally ruled out for offside.
Cameron Carter-Vickers thought he’d gifted the visitors the lead with an own goal, turning in Kyle Walker-Peters’ low cross but the defender was deemed marginally offside in the build-up.
Djenepo, however, pushed Saints in front after Redmond raced into the Bournemouth half before playing the winger on goal to curl an effort beyond Asmir Begovic.
Redmond then doubled the lead on the stroke of half-time with a bit of individual brilliance that ended with him striking a right-footed effort into the top corner.
Jack Wilshere had a chance to draw Bournemouth back into the cup tie three minutes after the restart, but hit the side netting from the edge of the box.
Southampton would push the game out of reach in the 59th minute when Redmond doubled his tally, turning in Armstrong’s rebound to make it 3-0.
Southampton, who have struggled in the league with just one since January 4, are looking to make the FA Cup for the first time since 2003.
The FA Cup semifinals will take place on April 17 at Wembley Stadium.
Bournemouth, meanwhile, return to league play against Middlesbrough on Friday and are sitting outside the playoff spots with 11 games to go.
Bournemouth (4-2-3-1): Begovic; Rico (Mepham 67), Cook, Carter-Vickers, Stacey; Pearson (Surridge 56), Wilshere; Danjuma, Biling (Kilkenny 77), Riquelme (Anthony 56), Solanke
Unused subs: Travers (GK), Zemura, Saydee, Moriah-Welsh
Bookings: Wilshere (84), Surridge (90+3)
Southampton (4-4-2): Forster; Bertrand, Vestergaard (Salisu 77), Bednarek (Stephens 77), Walker-Peters (Ramsay 89); Djenepo (Tella 85), Diallo, Ward-Prowse, Armstrong; Redmond, Adams (N’Lundulu 90)
Unused subs: McCarthy (GK), Ferry, Jankewitz, Watts
Goals: Djenepo (37), Redmond (45+1, 59)
Bookings: Bednarek (30), Vestergaard (74)