Chelsea captain Terry opened the scoring shortly before half-time as the Premier League leaders secured a first trophy since Jose Mourinho's return as manager in 2013.
Mourinho's side pulled further clear when Diego Costa's 56th-minute strike took a deflection off Kyle Walker to beat Hugo Lloris and Tottenham never seriously threatened to pull off a fightback thereafter.
"That's the first one," Terry told Sky Sports before being presented with the trophy.
"It's massive. That could be the start of something very good but now we need to kick on and we've got the league to focus on.
"I thought we handled the game very well and we're delighted with the win, obviously."
Terry felt Chelsea were "deserved winners" and praised Mourinho for taking the competition seriously.
"He made it very clear from the outset that he wanted to win this competition and we've done it, so credit to the manager and the squad," added the former England defender.
