Nigel Clough was sacked on Saturday evening just hours after the Rams lost 1-0 to east midlands rivals Nottingham Forest at the City Ground, bringing an end to his four-and-a-half year reign as manager.
By Monday afternoon a deal had been done to bring McClaren to the club from QPR, where he had been coaching, and his backroom staff had been assembled - with Paul Simpson joining as first-team coach and Eric Steele as goalkeeper coach.
Asked about the quick turnaround, Rush said: "Firstly, parting with Nigel was not a decision that was taken lightly and we are on record as thanking him for all the great work he has done here.
"But once that decision was made, I think it is important once you identify someone that you do it as quickly as you can.
"So yes, I spoke to Philip Beard at Queens Park Rangers on Saturday evening and asked him if I could have permission (to talk to McClaren) and we moved from there.
"Ultimately, from my point of view once you make the decision, you look to how you can move the process forward.
"Am I pleased we have got Steve McClaren, Eric Steele and Paul Simpson now moving forward with this club? Of course, I am delighted.
"What I didn't want to do was put a timeframe on it.
"You have your plans and in lots of ways it is no different to a transfer. Look at the Johnny Russell transfer in the summer. It was on, it was off, but other deals you can do in 48 hours and no-one knows about them.
"I am pleased it has turned out as it has, but others will be the judge of whether it is successful or not."