On a day when Premier League title rivals Manchester City were beaten at Anfield by Liverpool, goals either side of the break secured Mourinho's first silverware since returning to Stamford Bridge in 2013 on a hugely positive afternoon for Chelsea.
Spurs came from a goal behind to win the last League Cup final meeting between these teams in 2008, but even in-form Harry Kane - who struck twice in January's 5-3 league win over Chelsea - was unable to haul his side back into the contest after Terry's opener.
In a competitive first half that saw Christian Eriksen strike the crossbar with a curled free-kick and the feisty Costa heavily involved in a number of flashpoints, Terry's 45th-minute strike sent Chelsea in ahead at the break.
Costa then got on the scoresheet 11 minutes into the second half, courtesy of a hefty deflection off Kyle Walker, with Chelsea able to successfully shut out the likes of Kane despite Kurt Zouma having to slot into a makeshift midfield due to Nemanja Matic's much-discussed suspension and John Obi Mikel's injury.
In the heavy rain at Wembley, Mauricio Pochettino's hopes of ending Spurs' seven-year trophy drought drained away as Mourinho's side look ahead to Premier League and UEFA Champions League challenges.
Mourinho made five changes from Chelsea's 1-1 draw with Burnley and saw his side make the brighter start as Terry headed over Willian's cross before Branislav Ivanovic's looping cross threatened Spurs' goal.
Hugo Lloris - making his first League Cup appearance of the season - dealt with the danger prior to Eriksen rattling the woodwork.
The Dane's 25-yard set-piece came about after a foul on Kane and the Spurs striker was next to go close with a low strike that tested Petr Cech.
Willian's raking pass set Eden Hazard away down the left, although his strike was blocked by Eric Dier, before Costa was involved in two flashpoints prior to the half-hour mark, having raised his hands towards Nabil Bentaleb.
A knock to Walker set the tone for a poor finish to the half for Spurs that was compounded when Terry made the most of lax defending on the stroke of half-time.
When Chadli was caught out by a looping ball and fouled Ivanovic, a host of Spurs defenders failed to clear Willian's delivery, allowing skipper Terry to open the scoring from close range courtesy of a deflection off Kane.
Cesc Fabregas' overhead kick forced Lloris into a smart save shortly after the interval and Chelsea's positive start to the second half was rewarded when Costa scored his first goal since mid-January.
A purposeful burst down the left was helped past Lloris by a Walker deflection 56 minutes in, with Pochettino opting to introduce Mousa Dembele and Erik Lamela to try and gain a foothold in the contest.
However, Chelsea demonstrated their defensive solidity to see out the victory with minimal fuss and collect the first of three potential trophies this season, while Spurs must focus on securing European qualification.
