Lichaj opened the scoring for Forest in a stunning 4-2 victory over holders Arsenal in the third round before his spectacular volley made it 2-1 to the home side at the City Ground.
Forest's remaining two goals were each scored from the penalty spot, by Ben Brereton and Kieran Dowell, respectively, but Lichaj — who had never scored more than one goal in a match — revealed his added motivation to net a third.
"My wife told me if I get a hat-trick anytime during the year that I get a dog, so I was trying to get the penalty but [David] Vaughan told me politely — I don't want to say what he said to me... to go away," Lichaj told BT Sport.
Discussing his side's performance, caretaker manager Gary Brazil added: "That was a fantastic day for the football club. I may be a little bit biased but I think it was a well deserved win.
"There were some big performances from the players this afternoon and I always felt comfortable we would go on and win the game, and so it proved.
"It would have been foolish to play the game Arsenal wanted us to, so we had to go on the front foot and get in amongst them. We didn't want to give them space or time on the ball or allow them to dictate the game.
"Our first 10 minutes showed our intent and we really set the tempo of the contest."