The 25-year-old was denied a third goal in the 5-0 rout after Jonathan de Guzman refused to pass over penalty-taking duties, but he hailed the Welsh club's "massive" achievement.
After scoring a goal either side of halftime, Dyer had the opportunity to secure his third tally, but De Guzman declined to pass on the penalty-taking duties after a lengthy dispute and went on to calmly convert from the spot.
When asked what it was like to score twice in the final, Dyer told Sky Sports: "It was brilliant. I haven't scored for a while so to get two goals today, obviously I wanted the hat trick with the penalty.
"Obviously, I had to score on a historical moment like this, and I thought I should have gotten the ball, but it wasn't meant to be and I passed it on and that's the way it went."
Swansea's cup win is the first major trophy in club history, which also secured Europa League qualification for Michael Laudrup's team. Dyer praised the club's progress from the lower reaches of English football and hailed a "massive" day for the city as a whole.
"I think Swansea City have come so far as a club and a city," Dyer said. "For us to win this cup final is massive for us. The players have worked so hard all week, and it's just a great day."