After a tight opening period the visitors took the lead when Luke Chadwick, on loan with the Clarets from Premiership Champions Manchester United, capitalised on a slip by Matt Sparrow to put the ball past Sam Russell.
Despite the gulf in league positions between the two sides United fought back well, with Richard Kell going close before Steve McLean headed home the equaliser.
United needed to hold on until the break, but found themselves behind again when Robbie Blake beat his man before firing home his fifth of the season.
The second half saw Brian Laws' side pushing for the equaliser, with visiting full-back Dean West twice clearing Steve Torpey efforts off the line.
Ian Moore fired in an unstoppable third for the visitors, which looked to be enough to kill off any linering hopes of an Iron fightback.
The introduction of veteran winger Peter Beagrie added extra impetus, with Beags firing in a low free-kick five minutes from time to ensure a nervous finish.
Another equaliser failed to materialise, and although there were positives for Iron fans to take from the game it was the First Division club who go into the draw for the Third Round of the competition.
Iron lineup:
Russell, Sharp, McCombe, Byrne, Butler, Featherstone, Kell, Barwick, Sparrow, Torpey, McLean.
Burnley lineup:
Jensen, West, Camara, May, Todd, Weller, Farrelly. Blake, Moore, Facey, Chadwick.
