Despite sitting over a 100 places below Tsurenko in the rankings, Burnett, who was the last direct main draw entrant, looked confident from the start, breaking serve twice in the opening four games before overcoming a slight wobble to take the first set on a tie-break.
The second was another see-saw affair, with breaks - and double faults - aplenty. Once again, though, it was Burnett who maintained her composure when it mattered most, hanging on in a marathon final game to close out a 76(4) 64 win on her third match point.
"I was surprised that I was in the main draw at all and to win today is an even bigger surprise," Burnett said. "I came into the match with nothing to lose and was relaxed and played my game.
"In both sets I was dominating in the early part, but Tsurenko kept pressuring me and the match got closer. I was feeling a little nervous and was in a hurry to finish the match, but tried to just play my game and keep going for my shots and eventually clinched it."
Joining Burnett in the second round was last year's surprise package, Donna Vekic, who eased past Lyudmyla Kichenok, 63 62.
Twelve months ago, Vekic, then just 16, became the youngest WTA finalist in six years, and in the intervening period she has shown this was no flash in the pan with a rapid ascent up the rankings.