Terry has been Chelsea captain for all nine of their attempts to win the Champions League under billionaire owner Roman Abramovich, suffering four semi-final defeats and an agonising loss in the 2008 final when the defender missed a penalty to win the game.
He said: "Eight, nine years, I've been waiting for this.
"Being knocked out of competitions in the last minute, losing on penalties before in Moscow and losing against Liverpool, that's something that lived with me.
"But, all of a sudden, a big weight is off my shoulders and probably everyone within this football club."
Unlike Moscow, Terry was forced to watch Saturday night's penalty shootout victory from the sidelines thanks to his senseless semi-final sending-off.
"I can live with that, I really can, because the club have won," he said. "Things are just meant to be. And I wasn't meant to play.
"We've gone on to win it and, for me, that's all that matters because I care about this football club so much.
"And to see us win the trophy that I feel we deserve and to see the smile on the owner's face and those fans at the end, they deserve it."