The Englishman grabbed his second from the penalty spot with 19 minutes remaining.
Kevin Long then poked home to round off the victory.
The Clarets will face Bournemouth or Crawley Town in the next-round, with their match to be played on Tuesday.
Speaking after the game, Clarets boss Sean Dyche was positive about his team's chances in the tournament.
He said: "There were so many good performances from players who haven't played too much. I know they are good players but they remind you with displays like that.
"I'm shocked we got a penalty, I can't believe it! We average one every 24 games normally and that's two in a few days.
"We take each game as it comes, there are no guarantees. If we keep going as we have been lately who knows how far we can go? When everyone is fit I believe in all of them."