Anderson, 33, believed he was fit after a shoulder injury but the England selectors were unwilling to risk him.
"If we'd won, that story wouldn't have blown up like it has," said Cook.
England have trimmed pace bowlers Jake Ball and Steven Finn from their 14-man party ahead of second Test, which starts on Friday at Old Trafford.
Instead of playing for the national side last week Anderson, England's all-time leading Test wicket-taker with 454 dismissals, bowled 22 overs for Lancashire, taking 3-58 in their County Championship defeat by Durham at Southport.
"It happened, it was a slightly messy affair and an unimportant sideshow. We have to move on and play good cricket," added Cook.
Pakistan won by 75 runs at Lord's to take a 1-0 lead in the four-Test series.
