McBurnie, who notched two goals during the Championship clash, put in a strong performance against his hometown club in south Wales.
The 22-year-old was linked with a move to Elland Road throughout the summer but Leeds' interest never materialised as he returned to the Swans following a loan spell at Barnsley.
McBurnie though believes that Graham Potter's side sent out a message to the rest of the division with their hard-fought draw against the Whites, telling the BBC: "I guess everyone expected us to get pumped after the weekend and we've shut a lot of people up with that performance.
"Everyone thought Leeds were massive favourites going into the game, but they weren't favourites in our dressing room."
The striker, who is originally from Leeds, was released by United as a schoolboy for being too small and revealed that he had friends in the away section of the Liberty Stadium on Tuesday.
"They were all in the crowd," McBurnie continued.
"They all said they wanted me to score but Leeds to win so I would have been able to brag over them if we'd got the win.
"Me and my brother were both there when we were kids. Growing up in Leeds, it's nice to score against them, but the result is a bit disappointing.
"I got released for not being tall enough, believe it or not (he is now 6ft 3in). I was quite small when I was 15. I could never hold a grudge against them for that. I wouldn't be where I was today if that hadn't happened to me.
"It's made me a lot mentally stronger and stuff like that, so it's nice to score a couple against them."