Anderson showed admirable stamina as he bowled 13 successive overs on the final morning at Trent Bridge, his burst of three wickets for six runs helping to give him 10 in the match.
His final wicket was in keeping with a match littered with twists, turns and controversy, with a DRS referral needed before Brad Haddin (71) was ruled to have got a slight edge to Matt Prior behind the stumps.
"Australia fought incredibly hard and a lot of credit to them for the way those guys batted today, but we just hung in there incredibly well," Cook said at the post-match presentation.
"Jimmy was outstanding. He always wants one more over - I think 13 was probably quite a lot in that first hour!"
Asked if there was an over-reliance on Anderson, Cook said: "No, not at all. He's a world-class bowler and you sometimes use him in these situations when you know there's a timeframe. He had an amazing rhythm in this game.
"But [Stuart] Broad and [Steven] Finn have done outstandingly well for us over a huge amount of time, but it just happened to be Jimmy's day and Jimmy's game. Sometimes it happens like that."
Ian Bell's second-innings 109 also came in for praise from the skipper.
"It was a real innings of character, determination and skill," he said.
Anderson ended the contest with match figures of 10 for 158, and asked if it had been his best performance in an England shirt, he told Sky Sports: "Yeah I'd say so.
"We knew it wasn't going to be easy at the start of the day, but I thought the lads stuck to it and stayed calm when it was getting tough.
"I'm just delighted to get the win."
He added: "I had the nerves going a little bit but I love bowling here, it's been good to me over the years and I'm happy that I could pick up some more wickets."
Asked if he could maintain the same levels of workload across the series, he said: "If I'm bowling 13-over spells it's going to be difficult!
"But it's Ashes cricket. I love playing Test match cricket and the Ashes is right up there, and that's why we do the hard work in the gym - to get out there and bowl long spells."
Australia captain Michael Clarke, meanwhile, was proud of his players for their contribution to a memorable Test encounter.
"The boys can hold their heads high," he said. "It was a wonderful game of cricket but credit to England.
"They fought really hard through the five days and the two best performers were Ian Bell making a hundred and Jimmy Anderson getting five in both innings, so they deserved to win."
Ashton Agar's 98 on his Test debut was an obvious highlight for the Australians and one that bodes well for the rest of the series, and Clarke had special praise for the 19-year-old.
"He's played really well," he said. "He's an amazing talent, a great kid. He's got a very smart mind. He's a lot older than his age would tell you.
"He loves the game, he's enjoying being around the Australian family, and I think you're going to see a lot more of him."
Asked if Australia's 10th-wicket heroics in both innings reflected badly on the top order, Clarke said: "I don't care how we make the runs, so long as we get them.
"We're prepared well, we had a really good five days of cricket, plenty of ups and downs, and we can hold our heads high."
Reflecting on the rest of the series, which continues at Lord's on Thursday, he added: "I'm as confident as I was when we landed here.
"We get another crack in four days so we'll look forward to that."