On a potentially tricky afternoon for the Swans against League Two surprise package Bradford City, Dyer settled any nerves at Wembley as he opened the scoring on 16 minutes.
After Michu made it two, Dyer than netted again early in the second half to put Michael Laudrup's side firmly on course for a first major trophy triumph.
He could have taken home the match ball, but saw Jonathan de Guzman refuse to relinquish penalty-taking duties as the Swans swept to a comfortable 5-0 win.
Asked for his take on proceedings, Dyer told Sky Sports: "I haven't scored for a while, so to get two goals today...
"I obviously wanted the hat-trick with the penalty. I thought I should have got the ball, but it wasn't meant to be and I passed it on. That's the way it went."
On Swansea's achievement, he added: "Swansea City have come so far as a club. To win this cup final is massive for us. The players worked so hard, it's been a great day."
Swansea manager Michael Laudrup blamed himself for the dispute over who took the penalty.
"There was no designated penalty taker, I forgot," he said.
"This was our 36th game of the season and it was our first penalty, so it is my fault because I didn't say who was going to take it if we got one.
"I thought we would go through the whole season without getting one!"
Leon Britton echoed the sentiments of his team-mate Dyer and, as a man who has represented the Swans in all four divisions of the Football League, he was eager to point out how far the Welsh outfit have come.
He said: "It's a dream come true. As a kid you dream and watch all the cup finals. Today is a dream come true.
"It's been unbelievable, the last 10 years. I know we keep harping on about it, but we nearly went out of the Football League and here we are today in the Premier League and winning our first major honour and qualifying for Europe.
"It will be a great adventure next season for the players and the fans alike."