There was to be no repeat of last year's heroics from the Bantams, who caused shockwaves in last season's competition as they became the first fourth-tier side to reach the final in over half a century.
Vaughan, signed by Huddersfield for B£1million from Norwich this summer, scored either side of the break to end Bradford's involvement despite the League One newcomers pulling a goal back late on through substitute Nahki Wells.
"James did what he's paid to do. He's been champing at the bit to get on the end of something," Robins said. "His second goal in particular was good. It was a great cross from Adam Hammill but then James produced a powerful and controlled header.
"I'm keen to get the lads creating chances for him. But we need to get the balance between attack and defence right."
On the game, Robins added: "I enjoyed watching it. The boys played well and we took the lead at the right time.
"A few people were getting a bit restless, but we grew into the game and after that there were some really good performances.
"Bradford are a good side and have momentum behind them. And any team with Nakhi Wells and James Hanson is going to be a threat."
Robins singled out debutant full-back Jake Carroll for praise after the 21-year-old's summer move from Irish side St Patrick's Athletic.
He said: "He's going to be a really good player. He's got lots of potential and he showed it tonight. Jake didn't look out of place."