The Blades grabbed a shock 20th-minute lead through Jamie Murphy.
Villa struggled for much of the afternoon until substitute Nicklas Helenius hit a barely deserved equaliser on 75 minutes - only for Ryan Flynn to fire his superb winner past Jed Steer five minutes later.
Clough said: "I don't think he (Lambert) was disrespecting the FA Cup - he was just stating very honestly what the situation is like in the Premier League at the moment.
"It is just a bit sad that the FA Cup, such a wonderful competition,
takes a back seat but that is the reality of it.
"For us and our 6,000 fans who came today, I think you saw how much it meant. It's still a special, special competition and we'd just like a home draw next because we've had three away games on the spin so it would be nice to get one at Bramall Lane."
Clough added: "We had a great chance just after their equaliser. The equaliser gave us that kick up the a**e and we went and attacked them again and got a deserved second goal.
"We could have come here to try to sit back but we don't want to play like that."
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