Anderson bowled 13 straight overs before lunch as he finished with an innings return of five wickets for 73 runs and a match haul of 10 for 158, only the second time in his 83-Test career he's taken 10 or more wickets in a match.
A clash beset by umpiring rows and controversy over the Decision Review System ended shortly after lunch on the fifth and final day when England overturned umpire Aleem Dar's original not out decision to have Brad Haddin (71) caught behind off Anderson.
"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.
"Jimmy was outstanding. He always wants one more over -- I think 13 was probably quite a lot in that first hour!"
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Cook also highlighted Ian Bell's second innings 109, which helped give England just enough runs to defend, as a key contribution to the match.
"It was a real innings of character, determination and skill," he said.
Anderson's match figures were his best in Test cricket since he took 11 for 71 against Pakistan at Trent Bridge in 2010.
"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," Anderson said.