The result ended the visitors' 12-match unbeaten run and saw the Clarets knocked off the top of the table.
Robins said: "There's bound to be talk over the penalty.
"I haven't seen it but I was told it might have been outside the box. But that shouldn't be allowed to take the shine off what was an outstanding performance from us.
"I am really proud of the lads and even if we hadn't taken the three points I would have been delighted."
Burnley manager Sean Dyche felt the penalty should not have been given.
"I have seen it three or four times and it's outside the box and I don't think there was any real contact with the hand," he said.
"There's not much point in me talking about it because I will get into trouble.
"It's quite obvious it's not a penalty because it wasn't in the box - unless the rules have changed."
It was the Terriers' first win against a top-half side this season.
"If we keep doing that for a full 90 minutes we will be a tough team to play against," Robins said.
Huddersfield could have had the game won by half-time but Vaughan squandered a series of chances and strike partner Martin Paterson's header struck a post.
After a goalless first half, Town clinched victory with two goals in a five-minute spell.
Oliver Norwood gave the hosts the lead in the 50th minute with a training-ground routine.
Adam Hammill chipped up the corner for Norwood 30 yards out to strike on the volley. Goalkeeper Tom Heaton could only parry the volley and the ball ended up in the roof of the net.
"We work on things in training and when they come off it's great. We work our socks off in training," Robins said.
Five minutes later Duff conceded the penalty and referee Eddie Ilderton booked him for disputing the decision. Vaughan coolly slotted home the spot-kick.
Burnley fought back and Sam Vokes' effort was ruled out for offside before the Clarets grabbed a lifeline when Norwood's misdirected header was pounced on by Vokes, who set up Danny Ings to score six minutes from time.
Duff saw red for his second bookable offence in the fourth minute of stoppage time when he pulled back Vaughan.
Dyche was not sure whether Vokes was offside for the disallowed 'goal' and insisted the corner for the first goal should not have been awarded.
"We have had two big decisions go against us and that has also led to a sending-off, so it's a double whammy," he said.
"It's a real tough one for the referee. I believe in referees but it's frustrating for us and frustrating for the fans.
"I am always truthful about my team and we were second best in the first half. Huddersfield raised their game but that's what happens when you're at the top."
- Sports & Recreation
- James Vaughan
- Michael Duff