The Peterborough loanee scored either side of Aden Flint and Jay Emmanuel-Thomas strikes to level the scores at 2-2 before Nile Ranger clinched a dramatic winner with three minutes remaining.
In the battle of the Robins, the defeat leaves Sean O'Driscoll's men in the bottom four in League One, despite being one of the favourites for promotion.
Ajose got his first in the 20th minute when he angled a clinical finish past Frankie Fielding after a powerful run before he tapped in a second two minutes before half-time after a cross from Australian Massimo Luongo.
Cooper said of his midfielder's star performance. "Nicky has come in and scored two goals, he can do that.
"He has been at Manchester United a number of years which you can see, we are trying to rebuild his confidence. He has not played a lot of games for a while, and he's another one we are trying to rebuild."
"The game was too frantic for me, it was a fantastic game for the supporters to watch, I think if you pay money to watch that you won't have been short changed.
"It was a local derby and a real battle but when we pass and move the ball we are a threat. We have some really good players and I am pleased for the boys who came into the side today.
"I thought Louis Thompson was superb today when he came in, and Dany N'Guessan did well too."