He was on trial at the High Court in Aberdeen, alongside Martin Smart, 18, who was convicted of stealing the car.
Grampian Police Det Ch Insp Martin Dunn said: "Robert Wemyss showed a complete disregard for the safety of members of the public and police officers."
Wemyss was found guilty of car theft, dangerous driving and attempted murder.
Wemyss drove along footpaths and grass verges as he pursued the police through several streets in Aberdeen, eventually crashing into their unmarked car.
Other drivers were forced to take evasive action.
The BMW was driven at the three police officers as they escaped from their damaged vehicle and ran for cover.
It took a jury of eight women and seven men just under two hours to convict the pair.
Det Ch Insp Dunn added: "It is only by chance that his actions didn't lead to someone suffering serious injury or losing their life and he will now have to face the consequences of his despicable actions."
Sentence was deferred for reports until 15 March. Both men were remanded in custody.