The hosts took the lead in freak circumstances in the 63rd minute when Toffees goalkeeper Tim Howard, assisted by the blustery conditions, beat opposite number Adam Bogdan with a clearance from his own box.
Wanderers recovered from the blow swiftly, though, with David Ngog equalising four minutes later, before Cahill - once again named as captain despite a fee for him having been agreed with Chelsea - smashed home the decisive goal with 12 minutes remaining, securing a result which lifts the Trotters off the foot of the Barclays Premier League.
Coyle's decision to keep faith with the defender even though he looks set to leave the club proved a shrewd one and the Scot paid tribute to Cahill's mental strength.
"It can't be easy when it is a club of Chelsea's stature and a fee has been agreed," Coyle said.
"To be fair to them, I think they have respected the fact that we had a huge game tonight and have not then had further dialogue - we expect that they will possibly in the next few days.
"But while that continues, Gary shows tremendous mental character and desire for Bolton Wanderers, because he knows, as we know, how good a club this has been for him.
"That is why he will play at the highest level for years to come."
Coyle - who also named Cahill as skipper for the 1-1 draw with Wolves on New Year's Eve, 24 hours after revealing agreement on a fee had been reached with Chelsea - said there had still only been one meeting between Cahill's representatives and the London club, and was unable to say whether or not tonight's was likely to be the England international's last game for Wanderers.
"I was asked after the Wolves game as well whether it would be his last game for Bolton, and I can't answer that," said Coyle, whose team are up to 18th, a point adrift of safety.
"All I can tell you is what I know. Gary's representative has had one meeting with Chelsea and as with any negotiations, you have an opening gambit and then they obviously have to go back and speak to Mr (Roman) Abramovich (Chelsea's owner).
"I am led to believe that he has been away, so I think that is where we are.
"If Mr Abramovich sanctions whatever he wants to sanction, then they will take it from there."