The Toffees returned to the Premier League's top four with a 2-1 victory at the Liberty Stadium, sealed by Barkley's excellent 84th-minute set-piece.
It was another moment of magic from the 20-year-old who is enjoying a fine breakthrough season, and the England international admitted he always thought he would score.
"It's a relief because I had a chance to score with 20 minutes to go and I slipped, so to see the free-kick go in was a bonus and great for the team," he told Sky Sports.
"I told Romelu Lukaku to leave it, because I saw the goal and thought 'I'm going to score this'.
"I don't really practice free-kicks because Leighton Baines is usually on the pitch, so I thought it was my chance."
Skipper Phil Jagielka was happy that Barkley was on his side.
"He slipped and thought it wasn't his day, then five minutes later he hits the bar. Then he's brave enough to take the free-kick and he did," he said.
"It's a massive result because everyone around us won. It was a big game for us, we might have been fortunate to take all three points but that's the sign of a good team."