Ralph Hasenhuttl's men parked their Premier League problems as this competition once again provided a welcome distraction and shot in the arm for Saints.
Redmond starred in the clash of the south coast sides at Vitality Stadium, where he followed a smart assist for Moussa Djenepo's opener with a fine brace as 1976 winners Southampton secured a routine 3-0 win.
A trip to Wembley awaits in April after comfortably overcoming the Cherries, who never looked like progressing from just the second quarter-final in the Championship club's history.
There was early relief for Jonathan Woodgate's men when Cameron Carter-Vickers' own goal was ruled out by the video assistant referee, but Redmond slipped through Djenepo to put Saints ahead as half-time approached.
The forward followed his assist with an excellent solo goal in first half stoppage-time, then completed his brace and a comprehensive victory shortly after a Che Adams goal was ruled out by the VAR in the second period.
Hasenhuttl's men started with the style of a Premier League side in Dorset, making life uncomfortable for the hosts and forcing them into a 10th minute own goal.
Yet, without Danny Ings, the visitors were looking blunt and their increasingly sloppy play was giving the Cherries hope, with lively Arnaut Danjuma forcing Fraser Forster into a save at his near post.
Rodrigo Riquelme began to get more involved as the first half wore on, sending a first-time strike whizzing wide before aiming a low effort at Forster's goal.
But just as Saints began to wobble, things clicked moving forwards as they scored a stylish 37th minute opener.
Redmond collected a smart ball out from Walker-Peters on the halfway line and darted towards the Bournemouth box. Showing great composure and skill, he played a defence-splitting pass through for Djenepo to a curl a lovely shot past Asmir Begovic.
Hasenhuttl's men followed that blow with another in first half stoppage-time.
The forward took the Bournemouth captain wide, made him commit to a challenge and continued beyond him, before sending a superb shot from a difficult angle across Begovic.
Woodgate's men attempted to reduce the deficit in an encouraging start to the second half, with former England midfielder Jack Wilshere firing wide.
That attacking impetus was leaving gaps at the back, though, and Adams fired home superbly in the 53rd minute, only for the VAR to come to Bournemouth's rescue once more.
But Saints did not have to wait long for their third as their high press paid dividends as Bournemouth faltered when trying to play out from the back.
James Ward-Prowse picked off a loose Jack Stacey pass, leading to Armstrong hitting a shot off the post and Redmond impressively sliding home the rebound.
That 59th minute goal ended the quarter-final as a contest.
Begovic fumed at his team-mates after Armstrong was able to get a low shot away from 25 yards, but those calls fell on deaf ears as Redmond bent just wide shortly after.
Danjuma forced Forster into action on a rare Cherries voyage forwards, while substitute Sam Surridge was denied towards the end.