The Magpies twice surrendered their lead - eventually drawing 2-2 and facing a trip to South Yorkshire for a replay.
Speaking to the club's website after the game, boss Vermiglio said it was a day of mixed emotions.
"The feeling of being in the hat is one of excitement and one of optimism," he said. "But the overall feeling is if we could've just held on a little bit longer, we could have been here celebrating.
"But I'm proud of the performances the boys have put in tofay against a very good side.
"We've really given a good account of ourselves today and it gives us confidence. We know we can go toe to toe with a team like Doncaster.
"We'll dust ourselves down, rest and recouperate and we'll go again. The lads have been outstanding.
"This was an opportunity for them to stand up and be counted and I think they all did that."