Walker started both legs of Man City's semi-final tie against Real Madrid, but he had to settle for a place on the bench in Istanbul.
The England international watched on as Rodri scored the decisive goal in the 68th minute, guiding the ball past the despairing Andre Onana.
Walker replaced John Stones in the 82nd minute, and he helped his side see out the remainder of the contest to win the Champions League for the first time in the club's history.
Having previously won the Premier League and FA Cup, Man City also become just the second English side to win a treble.
Speaking to BT Sport after the game, Walker said: "I'm over the moon. I'm rarely speechless. My dream has come true, to achieve this with this club.
"I'm always going to be disappointed when I'm not playing but I'm 33. My example sets to the younger lads. I made a speech before they went out.
"This club means so much to me. To experience what I've experienced, I'm forever in debt. No pressure. You can't say Inter deserved it more than we did or we deserved it more than they did.
"We're both here by merit. There has to be a loser. Emotions played a part. The occasion is obviously going to get to you.
"I got a silver one last time so to get a gold one is fantastic. I'm living the dream. My mum and dad are in the stands.
"From where I come from in Sheffield it's not easy. I'm living the dream. My mum and dad are in the stands. From where I come from in Sheffield it's not easy.
Walker, who joined Man City from Tottenham in 2017, has won a Champions League, five Premier League titles, two FA Cups, four League Cups and two Community Shields during his six-year stay with the club.
The Englishman has made 39 competitive appearances in the 2022-23 campaign, including five in the Champions League.