England, chasing 227 to win a home Ashes series 4-0 for the first time, were 21 runs short of their target when bad light forced the players off four overs early and the match ended in a draw.
"It would have been nice to finish with a win but the rules and regulations are there for a reason," Cook said at the presentation ceremony after lifting the famous urn while red and white fireworks erupted around the ground to celebrate England's third Ashes win in a row.
"The umpires have strict guidelines and, if it was day three, we would have gone off. This morning our job was to try and make it as difficult as possible to stop Australia from pushing on for a win and to make our chances of victory easier.
"The crowds have been fantastic for all five tests and I'm privileged to captain this group of guys. Winning this series is the proudest moment of my life."
Australia captain Michael Clarke declared his side's second innings on 111 for six, setting England a challenging target in the hope of forcing a consolation victory for his team.
"We're here to try and win, credit to England they played well today," Clarke said. "It was worth a crack.
"We did our best to set up the game and hopefully we have given the fans something to enjoy. We got outplayed throughout the series and again we couldn't get over the line today.
"England deserve a lot of credit for the way they played. The result says they won 3-0 and it doesn't matter what I think. The last three tests, we have played a terrific brand of cricket.
"The whole squad has worked their backside off but we have to give credit to England and Alastair Cook."
England batsman Ian Bell was named his team's player of the series after scoring 562 runs, including three centuries, at an average of 62.44.
"On a personal note it's nice to contribute," Bell said. Going into the series I was low on scores - I kept getting caught in the covers - but to get hundreds helped the team win some test matches. As the series went on, I got better and better."
Bell made his Ashes debut in 2005 as a junior member of Michael Vaughan's England team who won back the urn.
"I don't think I was very good back in 2005 but I learnt some big lessons in that series," Bell added.
"I am a more senior player now and I am contributing more. It's been a tough battle and we're all looking forward to a break from test cricket for a while."
Fast bowler Ryan Harris was Australia's player of the series after taking 24 wickets despite missing the first test at Trent Bridge.
"I probably wasn't ready for the first test, but the guys who did play did well but I got the opportunity and took it and enjoyed it," Harris said.
"I came here wanting to play five tests but I said I wanted to be on the plane and not go early. Putting games together and getting overs in gives me confidence.
The next Ashes series starts on November 21 in Brisbane.
"I will go home, get the hamstring twinge sorted out and be ready when England come to Australia," Harris said.
"It has been good fun but it has been challenging and there have been some good battles."
(Editing by John Mehaffey)