Clarke opened the scoring in the 14th minute with a close-range finish and struck an 87th-minute winner after Franck Moussa's shot was parried by George Long.
Sandwiched in between Clarke's double-strike was Dave Kitson's headed equaliser from Barry Robson's corner after 72 minutes.
Robins said: "All in all it was a great win. I'm delighted with them because they worked really hard.
"In the first half I thought we were brilliant. In the second half we were not as good but we came out on top.
"Sheffield United are a really good side and they put you under unbelievable pressure with the balls they pump in and we dealt with it for most of the night.
"Then we got a little bit ragged and they put us under pressure so we just tried to tighten it up a bit.
"We just got a bit edgy and in all fairness it was because of the way they played. They put us under pressure and as you start to tire it can pay dividends for the opposition.
"Leon (Clarke) looked like he was tiring but he came up with the winning goal with a good little tuck in."
