Adam Clayton and Sean Scannell were on target for Huddersfield as the Terriers booked a date with Premier League Wigan in the fifth round.
And Lillis was delighted to keep the club's cup dreams alive.
"This was a fantastic evening for the club," he said. "The big thing was to get the lads into the dressing room at half-time and make sure they continued to believe in themselves. That was the big word - belief. We are very happy, of course we are. It is a great feeling."
The victory capped a successful day for Huddersfield, who earlier confirmed they had been given permission from Coventry to talk to Mark Robins about their vacant managerial position.
The Terriers made their move on Monday night as they seek a replacement for the sacked Simon Grayson, but saw that request initially rejected. The Sky Blues have now given their manager permission to talk to the npower Championship club, meaning Huddersfield's caretaker boss Mark Lillis may not be in his role much longer.
Asked about the identity of Grayson's long-term successor, he added: "I don't really know anything about the managerial situation, although I'm told the club have made an approach for someone. But we've got a job to do and it was a case of making sure the players focused on what they had to do.
"It is great to be in the fifth round of the FA Cup and at home, and it will be great to pass it on to the new man if he is here.
"We have had a lot of adversity since I came into the job three weeks ago and we had a young side out there, but they did us proud."
- Sports & Recreation