The Rams were up against it for large parts at Hillsborough but when Wednesday were unable to break the deadlock, Bamford showed them how to do it in style.
With 12 minutes remaining, the Chelsea loanee curled in a wicked effort from 20 yards to register his fifth goal in six games and earn a 1-0 success.
The victory keeps Derby's hopes of automatic promotion firmly intact as they moved to within two points of second-placed Burnley.
And McClaren knows the chances of earning a top-two spot will only be enhanced with players such as Bamford at his disposal.
"We have got players who can produce quality, there wasn't enough on the night but as the game went on it opened up and you can't calculate for a goal like that," the former England manager said.
"It was sheer quality and class to win what was a scrappy game.
"We nearly took him off, but you watch him training and he struggles to hit the big net and hits a lot of cars!
"I thought 'Here were go again' but this time he kept it down and it was a thunderbolt.
"He can frustrate you and sometimes flits in and out of games, but he is a match-winner and that's what he was tonight.
"He scores important goals, they're not just any goals. We're just delighted we have got him and we are very surprised how well he has done."
Had it not been for a resolute defensive showing the Rams would have left South Yorkshire pointless and McClaren concedes his side were fortunate.
"On another night we might have drawn that or lost it but this time it went our way," he added. "The players are making it work.
"Sheffield Wednesday did a very good job on us and frustrated us and stopped us playing.
"We had to defend a lot of balls into the box and two the centre-backs will have a headache, they've had to head that many balls. We had to defend well and we did."
Wednesday manager Stuart Gray admits it was Bamford's quality that separated the two sides.
The Owls were dominant throughout but could not get the breakthrough, with Derby keeper Lee Grant denying Michail Antonio on three occasions, and they were made to pay.
"I don't think my players deserved that, I just felt we didn't work Lee Grant enough," Gray said.
"It was a lot of effort, commitment and enthusiasm by both teams but that sheer bit of quality separates the sides.
"It's back-to-back defeats and we haven't scored, but we didn't get that bit of luck in front of goal. The most important touch in football is the last one.
"If anything we looked like the team that was going to win it. We probably deserved a point but nobody saves that goal.
"All you ask for is 100 per cent and we got that from every player, we have just got to lick our wounds and get on with it."