The Tykes fell behind to a Steven Fletcher strike but hit back 60 seconds later through Jon Macken and then in first-half stoppage time with a spectacular strike from Anderson De Silva.
Burnley fought back with a goal created by Fernando Guerrero and scored by Chris Eagles, but Hugo Colace scored a bullet header with 15 minutes remaining to settle the tie.
Barnsley have not won at Oakwell this season with one draw and three defeats to their name in the Championship.
But they have taken four points from nine since former Manchester United striker Robins replaced Simon Davey as manager.
After seeing off the Premier League new boys to reach round four, the Barnsley boss said: "There's a three-month period called the close season and you can't score or win during that time.
"People are still tagging on to that you have not won at home for six months.
"It's not that long but when you're talking about home form, you want to be winning games and tonight is proof can beat top teams on home soil."
Of the win, Robins said: "Burnley paid us a real good compliment because they played the majority of the side that played against Sunderland and beat them at the weekend.
"And they made it clear right from the offset that they were going to come and attack us.
"That suited us and I brought Jonathan Macken in tonight and he was fantastic and did everything you need your centre forward to do to enable you to play.
"If you look at the great goals we have scored, there were some great balls into the box, there's a lot to be pleased with."
The last time the Tykes made it to the fourth round was in October 1999 when they also knocked out a Premier League side - Bradford.
Asked who he would like to see his side face next, he added: "Anybody at home. At this stage you want to play your cup ties at home. If we can make some money along the way then fantastic because it will strengthen our position in the market."
Burnley boss Owen Coyle, whose side enjoyed a famous run to the semi-finals of the same competition last season, was clearly disappointed with his side's exit.
After seeing his side beat Fulham, Chelsea and Arsenal and compete so valiantly with Tottenham last season, the Clarets boss was looking for his side to embark on another cup run.
"We came here to win," he said. "And I know we made six changes to the team from Saturday and including the substitutes we made nine changes.
"But the team was strong enough to win the game and I think we offered enough to do so.
"You have got to pay credit to Barnsley, any neutral watching it will have seen some good football, end to end, but I'm disappointed to have lost."