This was the Tykes' first win over United in nine attempts.
Barnsley are in the midst of their best run for a year and the victory extended their unbeaten record to five npower English Championship matches, which Robins believes is just reward for his side's recent performances.
"It has been a long time in coming," he said after the game. "I think it had been 11 years since we last beat them and we were good value for it.
"They set their stall out today to make it difficult for us. Gary [Speed]'s pre-match comments were it would be a war of attrition and they set they their stall out for it to be a game like that.
"For us, it gave us a little bit of a route to try and pass the ball and we got out pretty well.
"To switch the play like we did and pass like we did in a local derby game when you're under that kind of pressure I thought was testament to us.
"The only thing that let us down, I thought, was adding to the first goal.
"[It was good to see] Hugo breaking into the penalty area, which was something he used to do last season and what we've been missing this campaign.
"He's done that today and made up for the times this season where he hasn't actually ventured that far up the field.
"I think it was a well-worked goal and then decision making let us down in increasing the lead.
"But for us, for the supporters - who sang their hearts out today - it's another derby win for us and we're delighted.
"And the part the supporters play cannot be underestimated. I think it's a vital thing at home."