South Australia 240 and 1 for 89 trail Queensland 342 (Burns 85, Clayton 67, Bazley 64, McAndrew 5-93) by 13 runs
Burns was run out for 85 on day three of the Shield clash in Adelaide, before the Bulls slumped from 2 for 184 to be all out for 342 in reply to South Australia's 240.
The Redbacks then went to stumps at 1 for 89, still trailing by 13 but with a draw now by far the most likely option on the final day on Wednesday.
But after the first session on Tuesday it looked as if Queensland were going to be able to push for a big first-innings lead and outright result.
With Burns and Jack Clayton (67) at the crease, the pair had put on a 115-run stand to set up the possibility of a lead of close to 200.
Then eyeing his first Shield ton in 21 months, Burns set off for a quick single after going back to a ball and pushing it onto the offside.
Jake Carder then swooped on the ball, picked it up with one hand and threw down the bowlers' end stumps in one motion to run the former Test opener out.
"It sucked. I couldn't believe I ran myself out," Burns said. "I felt like an idiot walking off the field. Very frustrating. At lunch we were really happy with where we were.
"Obviously it didn't go to plan in the second session, it probably started with a run out which is very regrettable."
After claiming Sam Truloff's scalp on Monday evening, he edged off Sam Heazlett and Jimmy Pearson after lunch on Tuesday cheaply.
He then got the key wicket of Max Bryant cutting at gully, with a ball that angled in at the right-hander's body and left him with little room.
The right-armer's five-wicket haul was then complete when Xavier Bartlett pulled him to deep square leg, just balls after the quick copped a nasty blow to the ankle.
"It's nice to be back contributing with the ball," McAndrew said. "It's been a rocky start for me with the ball, just struggling with injuries. So it is nice to get that rhythm back and feel strong at the crease and like I am contributing to the boys."