The Aussies resumed day three at 0-282, in response to Zimbabwe's 300, and took a first innings lead of 198 in Harare, mainly thanks to Warner's highest ever first-class score of 211.
A couple of late wickets had the hosts at 3-49 and on the back foot having already lost the opening four-day match.
Warner and Finch (122) resumed their terrific opening partnership on day three, but only managed to add seven runs.
Finch was bowled by Ray Price (4-116) without adding to his overnight score and Usman Khawaja's poor tour continued when he was caught by Brendan Taylor off the bowling of Chris Mpofu (3-102) for a duck.
In his four innings in Zimbabwe, Khawaja has managed just 52 runs.
Warner's blistering innings continued when Callum Ferguson (48) came to the crease and the pair put on 78 for the third wicket.
But Elton Chigumbura (1-94) had Ferguson caught behind before captain Tim Paine (6) fell soon after.
When Mitchell Marsh (7) was clean bowled by Mpofu Australia A had been reduced to 5-400 after their remarkable start.
Warner, whose knock lasted 345 balls, featured 21 fours and two sixes, was the next to go - caught in front by Mpofu.
The New South Welshman had bettered his previous best first-class score of 114.
When Luke Butterworth (1) and Trent Copeland (0) fell to Brian Vitori (2-87) in the same over, the Aussies had lost 4-1.
But some lower-order hitting from Jason Krejza (51), Mitchell Starc (33) and Peter Siddle (13) helped the lead to a strong 198.
Zimbabwe made a shocking start to their second innings as Starc (2-8) struck twice in his first three overs.
First he had Vusi Sibanda (1) caught by Butterworth before having Zimbabwe's other opener - Tino Mawoyo (4) - edging to Paine.
Hamilton Masakadza (12) added 35 with Taylor (unbeaten 32) before Copeland (1-5) took his wicket.
Taylor, who made 85 in the first innings, will resume at the crease with Price (0).