Rudiger’s first half header ensured Chelsea pressed home the advantage they had earned with a 2-0 win in the first leg last week.
Thomas Tuchel’s team also had VAR official Mike Dean on their side as Tottenham were given two penalties by Andre Marriner, only for reviews to overturn both, while Harry Kane saw a goal disallowed for offside.
Chelsea will play Liverpool or Arsenal in the final at Wembley on February 27.
Liverpool host the coronavirus-delayed first leg of their semi-final against the Gunners on Thursday.
It is Chelsea’s first League Cup final appearance since 2019 as they aim to win the competition for the first time since 2015.
Tuchel has now taken Chelsea to three finals since taking over from Frank Lampard in January 2020.
They lost to Leicester in the FA Cup final before beating Manchester City to win the Champions League.
Securing a Wembley berth was the ideal preparation for Chelsea’s crucial date with Premier League leaders City on Saturday.
Tottenham have not won a major trophy since lifting the League Cup in 2008 and their options for ending that drought this season are dwindling.
Antonio Conte had conceded it would be hard to overturn the deficit against a team of Chelsea’s quality.
And the Tottenham manager’s pessimism proved well founded as Chelsea quickly put a stranglehold on the tie in north London.
Romelu Lukaku almost bagged an early goal when the Chelsea striker barged past Ben Davies for a strike that was saved by Pierluigi Gollini, with the rebound off Japhet Tanganga drifting just wide.
Tottenham’s bid to become just the second team to reach the League Cup final after losing the first leg by more than one goal was shattered in the 18th minute.
Gollini was a surprise starter for Tottenham instead of Hugo Lloris, a Conte gamble that backfired as the keeper was at fault for Chelsea’s opener.
It was a shambolic goal to concede as Gollini came off his line in a failed attempt to punch clear from Mason Mount’s inswinging corner, allowing the unmarked Rudiger to guide his header into the top corner via the underside of the bar.
Rudiger had already scored in Chelsea’s Premier League win at Tottenham this term and once again the German defender’s aggression was too much for the north Londoners to handle.
Callum Hudson-Odoi had a chance to double Chelsea’s lead with a low shot from inside the area, but this time Gollini was equal to the threat.
Chelsea were dominating possession but Pierre-Emile Hojbjerg almost grabbed a lifeline for Tottenham when his deflected effort flashed just wide.
Tottenham thought they had won a penalty when Rudiger sent Hojbjerg tumbling with a sliding tackle on the Dane, but VAR changed Marriner’s decision to a free-kick as the tackle was clearly made outside the area.
Giovani Lo Celso wasted the set-piece and Tottenham were booed off at half-time as their Wembley hopes faded.
Marriner’s poor decisions were only adding to Tottenham’s frustration and the referee was forced to deny the hosts another penalty in the second half.
Kane’s pass sent Lucas Moura through on goal, but Chelsea keeper Kepa Arrizabalaga made a well-timed block with his legs and, after Marriner errantly pointed to the spot, VAR intervened to change the decision.
Kepa made another important save moments later, leaping to his left to claw away Emerson Royal’s header.
Incredibly, Tottenham endured a third VAR reversal when Kane swept his shot into the far corner in the 63rd minute, only for the review to show he was inches offside.