Brighton, who reached the quarter-finals last season, had won just two games away from home in all competitions before today but saw off the Cherries with a strong, attacking performance, which saw goals scored by Anthony Knockaert, Yves Bissouma and Florin Andone.
Chances were limited in a quiet start to the game, but Ryan Fraser missed a golden opportunity to test Jason Steele after 25 minutes, with the Scotsman missing the ball 15 yards out following a dangerous cross from Nathaniel Clyne, who was making his first appearance for Bournemouth after signing on loan for the club on Friday.
Clyne continued to be a threat down the right side and his new side somehow didn't take the lead after 28 minutes with Lys Mousset blazing over from around eight yards out despite having an empty net to aim at after Jordon Ibe's shot was blocked by Shane Duffy.
It was a huge let off for Brighton, but they took full advantage with Knockaert expertly driving home the opener from a clever Jurgen Locadia backheel following a fast break from the visitors — their first meaningful attack of the game.
In similar fashion to Wednesday night's trip to West Ham, Albion scored a second in quick succession to take control of the tie, with Bissouma driving forward and firing home off the post from 25 yards on the 34th minute.
The fantastic strike was the 22-year-olds first goal in a Brighton shirt.
Bournemouth pushed forward in search of a way back into the game, but continued to be vulnerable on the counter attack with Locadia's glancing header from Gaetan Bong's cross hitting the post after 40 minutes.
Bournemouth brought on David Brooks for teenager Kyle Taylor at half time in the hope it would inspire the hosts to a come back, and the Welshman was straight into the action — albeit it being a confrontation with Florin Andone.
The sub did pay dividends though, with the confident Brooks orchestrating a more efficient attacking performance, and starting a move which culminated in Marc Pugh rifling home from 20 yards to bring Bournemouth back into the game ten minutes after the restart.
Howe's men grew in confidence and very nearly found a quick fire equaliser with Jordon Ibe twisting and turning his way into the box, before his blocked shot landed at the feet of Ryan Fraser, whose dangerous flash cross wasn't met by anyone.
Brighton made the home side pay yet again though, with Shane Duffy brilliantly heading Knockaert's corner back across goal for Andone, who gratefully nodded home from six yards to restore their two-goal advantage after 64 minutes.
The goal — from Albion's first corner of the match, was Andone's third goal for the club and the 15th that Bournemouth have conceded in their last four games.
It was also a much needed goal for Brighton to prevent them squandering a two-goal away lead for the second time in a week.
Bournemouth didn't give up though, with Steele having to push away a good 20-yard strike from Brooks, before Pugh curled one just wide from around the same position on 80 minutes.
However, Chris Hughton's side held firm to clinch a much deserved victory.
The Cherries came out on top with a 2-0 victory the last time the sides met, just two weeks ago in the league, but Brighton are now unbeaten in their four games since then.
Bournemouth, meanwhile, remain without a win since beating Brighton in the league.
Brighton: Steele, Bruno, Bong, Duffy, Stephens, Kayal, Bissouma, Locadia, Andone, Knockaert, Balogun
Bournemouth: Boruc, Clyne, Simpson, S Cook, Rico, Pugh, Taylor, Surman, Ibe, Fraser, Mousset