The hosts are 1-33 needing a further 322 for victory on the final day at Allan Border Field on Thursday.
The visitors overturned a first-innings deficit to set the imposing total thanks to a blazing unbeaten century from Kuggeleijn, who had taken five wickets in Australia A's first innings to turn the tide.
The No.8 struck an 85-ball ton that featured four sixes, Kuggeleijn unbeaten on 101 when the last wicket fell at 468.
Sean Solia compiled an attractive 91 while Tom Bruce (51) and Muhammad Abbas (55) also offered valuable support.
It was a remarkable turnaround after they were skittled for 147 on day one, spinner Mitchell Swepson (0-119) experiencing the full spectrum of fortunes after taking 3-19 in less than four overs in the first innings.
Mark Steketee and Mitchell Perry both took four wickets each, as Cameron Bancroft and Caleb Jewell will resume the chase with Tim Ward the man out for 11.
"Obviously Kuggeleijn came in and struck the ball beautifully - that happens in cricket, but I think all of us guys were pretty happy with how we went about it," Perry said.
"Over 300 is a big chase on any final day and I think the wicket is going to start to play a few more tricks.
"Hopefully the boys can dig in in the first hour and set up a nice little finale late in the day."
The unofficial Test is one of two between the teams in Queensland, before three 50-over games.
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