The 26-year-old scored in the final minute of extra time to earn the Blades a fifth-round tie against either Nottingham Forest or Preston, and end a run of four games without a win.
"It is unbelievable. I am delighted to get the goal for my little son who was born a few weeks ago," Miller told ITV Sport.
"I am made up for him. We would much rather have those four wins in the cup as 12 points in the league, but we have to keep going and take this mentality into the league."
Miller is in his second season at Sheffield United, having joined from Crewe Alexandra in July 2012.