The Shrimps had beaten Coventry in the previous round and again they showed little fear against opponents two divisions higher.
The Lancashire side took a shock lead on 11 minutes.
Mark Duffy weaved an opening in the box and fed the ball through to Phil Jevons who stroked home his first goal of the season past Lee Grant from eight yards out.
Jevons then saw a second effort cleared off the line before the Clarets came back strongly.
Jay Rodriguez went close with a header before Brian Laws' team equalised on the half hour when Chris Eagles fired home Steve Thompson's lay off.
Morecambe started the second half on top with Laurence Wilson rattling the bar with a superb free kick.
But Burnley hit back and earned a penalty on the hour when Darren Moss brought down Eagles and Steve Thompson made no mistake from the spot.
Burnley added a third 17 minutes form time when Kevin McDonald's long range shot took a wicked deflection to beat Barry Roche.