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Queensland 2 for 88 (Burns 49*, McAndrew 1-18) trail South Australia 240 (Nielsen 90, McAndrew 46*, Barlett 4-64) by 152 runs
Neilsen saved South Australia from a horror start to post a first innings total of 240, Burns went to stumps on day two on 49 not out with Queensland 2 for 88 in reply.
The right-hander looked in complete control on a difficult Adelaide Oval wicket, batting through an extended final session of a day regularly interrupted by rain.
Without a substantial score in Queensland's past three games, Burns played nicely all around the ground in a crucial last three hours of play.
It came as he and Bryce Street put on 52 for the first wicket, before Thomas Kelly took one of the catches of the summer to remove Sam Truloff late.
After Truloff went back to Nathan McAndrew, Kelly reacted by diving to his left and taking the ball one-handed to send the right-hander on his way for 15.
Earlier, Queensland quick Xavier Bartlett had continued his fine start to the match when he had Nielsen caught behind for 90 in the key wicket of the Redbacks' innings.
After Bartlett took three wickets early on day one to leave the hosts 5 for 25, Nielsen was the man to stand up and lead the fightback for South Australia.
Nielsen combined for 60-run stands with Ben Manenti and McAndrew for the seventh and eighth wickets, with McAndrew finishing 46 not out, before edging Bartlett just after lunch on Monday.
Bartlett's effort came with Queensland missing Mark Steketee and Michael Neser to Prime Minister's XI duties, with both teams also without their Australian Test and one-day stars.
"With the likes of Nes and Stek out, it was nice to chip in and get a few early ones for the boys," Bartlett said after his 4 for 64.
"A theme for us bowling first this year, the new ball has been really crucial.
"There's still a lot of time in this game and a lot of work to be done for us, but we're in a good position now."
South Australia are without a win after four rounds of the Shield, while Queensland sit second with two wins, a draw and a loss.
McSweeney and Lehmann leave Queensland ruing missed opportunity
The visitors struck early on the final day but couldn't push further for victory
'It sucked' - Burns misses landmark hundred as South Australia mount fightback
James Bazley helped give Queensland a lead but it could have been more while Nathan McAndrew bagged five
Queensland in control after Nielsen holds SA together
Wicketkeeper made 90 out of 240 while Queensland's Xavier Bartlett took 4 for 64, before Joe Burns made 49 not out to see the visitors to stumps two down