Trailing 2-1 from the first leg in Gibraltar Paul Heatley levelled thing up on 33 minutes after Burns had been denied by keeper Matthew Silva.
But within five minutes the visitors had restored their one-goal advantage through Kyle Casciaro.
The Crues pushed for an equaliser but had to wait until 10 minutes from time as substitute Jude Winchester restored parity.
Deep in stoppage time, with the game seemingly heading towards extra time, up popped Burns with a spectacular winner to set up the mouth-watering clash with Basel.
“Looking forward to it is an understatement because I think we’re going out there for a very tough game, but that’s what you dream of.
“When you’re a wee boy you dream about playing against these big clubs, and teams,” Burns told the club’s Facebook page.
“We’ll go out there and love every minute of it, but it will be the same as Wolves we’ll do our job, they scored late on to beat us 2-0.
“We know what we’re up against, we’re not silly, but we’ll give our best.”
Jordan Owens started the game, and even though he didn’t get on the score sheet Burns was full of praise for the experienced forward.
“We struggled out there because the conditions, the heat and everything sort of went against us,” he said.
“But we knew that if we stayed in the tie we would have a chance once we got back to Seaview.
“Starting with big Chicken we just thought get it up to him and let him do what he does best.
“So many teams over the years have struggled against him when they’ve come here.
“We knew to get it to him quickly and as early as we could and if he gave us 45 minutes or an hour, whatever could.
“The big man got hurt just going into half time, but we couldn’t have asked for any more from him.
“He’s a big legend and he gave everything for the team.”