The Ashes-holders went into the match on the back of wins by 14 and 347 runs in the first two Tests at Trent Bridge and Lord's respectively.
But it was a different story at Old Trafford where they were under sustained pressure for the first time in the series.
Set 332 to win, after Australia captain Michael Clarke declared on his side's overnight 172 for seven, England collapsed to 37 for three on Monday's final day in the 94 minutes' play possible before rain saw the match abandoned as a draw.
That left England 2-0 up with only two Tests to play and, as the holders, they retain the Ashes.
"It's obviously a great feeling if a slightly strange feeling today," Cook said. "We were behind the eight ball for quite a lot of this game, but we fought hard today and had a little bit of luck with the weather.
"It's certainly nice to retain the Ashes."
"I think it's been a really good series," Cook added. "At Lord's we played really well. Trent Bridge was a nailbiting game and we were glad to get on the right side of that result."
England fell behind at Old Trafford as Australia declared on 527 for seven in their first innings with Clarke posting a man-of-the-match-winning 187 having won the toss.
"This game, it was an important toss to win but fair play to Australia, the way they played put us under a lot of pressure, getting 500 and plenty in that first innings, but we responded pretty well with the bat and fought hard," Cook said of an England first innings 368 featuring Kevin Pietersen's 113.
"Obviously the weather hasn't been ideal today but you can't predict that. It hasn't quite got the atmosphere because of that, but the feeling in the dressing room now is a very pleasant one," Cook, out for nought Monday, added.
"We set out to retain the Ashes and we've done that. After three Test matches I would have snapped your hand off to be in this position," he said. "Now we want to go and win the series."
For Clarke, winning the match award was no consolation for seeing the chance of regaining the Ashes washed away.
"I don't want to take anything away from England, they played good cricket and deserved to be 2-0 up," he said.
"That's the price you pay if you go 2-0 down in England, there's always a chance of a bit of rain and we've paid the price.
"But I'm proud of the way we've fought in this Test match."
As for his century, Clarke said: "I'd rather make a duck and have won the Test match."
England coach Andy Flower added: "I think the day for everyone was a bit of an anti-climax.
"It was a hard-fought Test match with two good teams going at each other, and a really excellent Test match pitch and the ground staff have to be complimented on that.
"I thought what really ensured our draw in this Test was the fight we showed in the first innings, to bat for 140 overs on a pitch already showing wear was tremendous."