First-half goals from captain Graham Alexander and Steven Fletcher put the Premier League side on course for only a second away win of the season and Coyle also saw his side hit the woodwork twice after the break before substitute Dean Morgan struck late on for the Dons.
Bolton have yet to approach Coyle about the vacancy at the Reebok Stadium but the former Wanderers striker remains a clear favourite with the bookies.
While Burnley find themselves currently two points better off in the table than Bolton, the Trotters' long-standing position in the top flight would give Coyle more financial clout.
However he would be leaving behind a team which he has moulded into an attractive and entertaining unit and the challenge to convert Bolton into a similar side may prove too much to ask.
For the Dons, the tie marked the start of a decade which could see them play a part in a World Cup in 2018 and they remain in the race for promotion themselves.
Manager Paul Ince won the FA Cup twice as a Manchester United player and saw his side compete well before Fletcher won a 23rd-minute penalty.
Fletcher cut in from the right byline and after beating one defender was tripped by Mathias Doumbe, leaving referee Tony Bates little option but to point to the spot.
Alexander stepped up and was his usual reliable self, smashing home off the right-hand post to give the Clarets the lead.
Robbie Blake came close to a second just after the half-hour mark after more good work from Fletcher.
Fletcher won a header to find Blake in space on the left of the area, and after a lovely turn deceived Luke Howell he sent in a shot which was well blocked by a sprawling Dons defender.
Burnley had kicked into gear and the impressive Fletcher doubled the lead in the 35th minute.
Chris Eagles seized on a loose Doumbe pass in midfield and slid Fletcher in on goal with a fabulous through-ball.
Fletcher cut inside onto his left foot and arrowed a low shot through goalkeeper Willy Gueret and into the back of the net.
Burnley started the second period in impressive style and came close to a third goal when Wade Elliott struck the crossbar.
A Blake free-kick hit the wall and Elliott smashed the loose ball past Gueret and against the woodwork.
Fletcher headed a free-kick onto the roof of the net and should have sealed the win after great play by Eagles.
Eagles tormented Dons captain Dean Lewington before picking out Fletcher with a perfect cross which the Scot headed powerfully past Gueret - only to see the ball strike the post.
The sizeable away fans were busy singing Coyle's name - as well as disparaging that of near neighbours Bolton - as the half progressed before the hosts came to life.
Peter Leven and substitute Luke Chadwick were both denied by goalkeeper Brian Jensen as Ince's side threw caution to the wind.
Jude Stirling was thrown on with 10 minutes to go and saw a volley with his first touch well blocked as the home side piled on the pressure.
And Stirling was the provider in the 89th minute when his cross was smashed home by substitute Morgan off the underside of the crossbar.
But Burnley held on to give Coyle a rare win on the road and join Bolton in the hat for round four.