"I am delighted with the players' performances and their attitude.
"The squad has knuckled down and when it was time to deliver, we did. We should not be afraid of anybody."
The victory took England above Russia in Group E - to sit three points behind leaders Croatia - with three games remaining, including a tough trip to Moscow.
And McClaren was quick to point out that England still have much to do before qualification is assured.
He added: "It is going to be a tough game in Russia, that is going to be the key match. But we still have more games left and have not qualified yet.
"However, the last two performances are encouraging - it has put down a marker for everyone else."