Sean Longstaff’s first-half double ensured Howe’s side finished the job after winning 1-0 in the first leg last week.
Che Adams reduced the deficit before the interval and Magpies midfielder Bruno Guimaraes was sent off in the closing stages.
But Newcastle held on to clinch a 3-1 aggregate victory that booked a final date with Manchester United or Nottingham Forest at Wembley on Feb 26.
United hold a 3-0 lead against Forest heading into tonight’s second leg at Old Trafford.
The Magpies’ most recent domestic final ended in defeat against United in the 1999 FA Cup.
“You want to be in finals of competitions to increase your status and make yourself more desirable for people to join. This is a club on the up,” Howe said.
“It was an intense game. At 2-0, we were playing really well. We are going for everything. As much as we can achieve, we will go for.”
Newcastle, who have never won the League Cup, are aiming to lift a major domestic trophy for the first time since the 1955 FA Cup.
Their last major silverware in any competition came in the 1969 Inter-Cities Fairs Cup.
On a cathartic night for Newcastle after decades of suffering, it was fitting that Longstaff, a boyhood Magpies fan, should be the one to send them to Wembley.
With Newcastle legends Alan Shearer and David Ginola joining the celebrations in the stands, raucous Magpies fans were ready to keep the party going all night long in the football-crazy city’s Bigg Market and Quayside bars.
Revitalised by Howe’s astute leadership and the financial muscle of a largely Saudi-backed ownership group, a club once regarded as the laughing stock of the Premier League can finally hold their heads high.
They sit third in the Premier League and look set for a sustained period as contenders for the English game’s top honours.
Reaching their first domestic cup final this century is another significant landmark in Newcastle’s rebirth.
“It is amazing. If you’d have said to anyone 12 months ago what was going to happen, they would have laughed,” Longstaff said.
“Since the takeover we have brought in quality players. It is really special and emotional for me what we are building here.”
Black and white wave
Ramping up an already vociferous atmosphere, Newcastle’s Anthony Gordon was paraded on the pitch before kick-off after completing his £45 million (B1.8 billion) move from Everton.
Howe had labelled the second leg a “season defining” moment for Newcastle and it took his fired-up side just five minutes to raise the decibel levels even further.
Longstaff found Kieran Trippier on the right flank and sprinted onto the defender’s return pass before guiding a low finish past Gavin Bazunu from 12 yards.
Newcastle had all the momentum, sweeping forward in a black and white wave that left Southampton powerless to stem the tide.
Once again it was Longstaff who provided the finishing touch to a breathtaking raid in the 21st minute.
Exposing Southampton’s fragile wing-back system, Joe Willock sprinted into space behind James Bree and found Miguel Almiron, whose cutback was drilled home by Longstaff.
Newcastle keeper Nick Pope had kept 10 successive clean-sheets, but his unbeaten streak ended in the 29th minute.
Completely against the run of play, Willock gifted possession to Adams and the striker smashed a superb shot into the far corner from outside the penalty area.
In the past, Newcastle would have wilted when Southampton pushed hard in a tense second half, but they are made of sterner stuff now.
Pope’s brilliant save from Adam Armstrong ensured Newcastle’s long-awaited return to Wembley wasn’t derailed despite Guimaraes’ 82nd minute red card for stamping on Samuel Edozie.
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