For the second successive round in the competition, the Millers prevailed on penalties in a shoot-out at the AESSEAL New York Stadium in the knock-out competition for lower-division clubs.
Thirteen spot-kicks were converted before U's player Adam May saw his effort saved by Viktor Johansson as Rotherham followed up their shoot-out success over visiting Port Vale in the previous round with a 7-6 triumph over Cambridge to book their place in the semi-finals. The draw is on Saturday.
One game from a return to the home of football, the Millers - who last won this competition in 1996 - also extended their unbeaten sequence on home soil to 16 matches in all competitions.
In normal time, the game ended deadlocked with Harding putting United in front with the first goal of his professional career early on.
But Cambridge, who produced an outstanding giant-killing feat in the FA Cup earlier this month by triumphing at Newcastle United, levelled thanks to well-taken 41st-minute goal from Sheffield born Paul Digby, who used to play for Barnsley.
On his side's penalty progress and a milestone goal for Harding, Warne said: I'll buy him (Harding) a cake. It's a huge moment for him. He caught it really well.
"I don't understand what's happened to football. All the lads can take really good penalties. It never used to be like that. You used to get two strikers who could take them and then it was the idiots.
"I have mixed emotions about the game. I thought we were really good in the first half and could have been out of sight.
"The first 20 minutes of the second half, I thought we were in complete control, but didn't create enough clear-cut chances.
"The last 25 minutes, Cambridge were probably the better side without creating anything. They controlled the middle of the park, which didn't please me. I made the subs to try to make us better and I don't think they did. I think we might have just edged it overall."