The Australian left-armer finished the 381-run win with nine wickets as players at both ends of England’s order struggled to deal with his pace and bounce.
The tourists were dismissed for 136 and 179 in their two innings and Cook knows that must improve if they are to have any chance of retaining the urn.
“Obviously he’s bowled very well and put us under some pressure which is something that we need to work on as a side going into the Adelaide game,” he said in the post-match presentation televised on Sky Sports 2.
“We’ve faced a lot of him in the past and been successful as well.
“There are times when he’s got on top of us and there are times when we’ve certainly had good times against him as well and we’re going to have to look to that for stuff to build on for Adelaide.”
Johnson, who became a figure of fun for England supporters in the last two series, also contributed with the bat as Australia recovered from 132 for six in their first innings, and Cook was left to rue letting the home side off the hook.
“Obviously it’s been a tough couple of days for us,” he said. “We got ourselves in a really good position there in that first innings.
“Credit to the way that (Brad) Haddin and Johnson played, they got Australia to a fairly competitive total but I thought it was still short (of par) on that wicket.
“We got off to a reasonable start with the batting and that collapse really put us behind.
“You always talk about trying to not lose wickets in clusters. You talk about it a lot of the time but sometimes that doesn’t really help you to do that out in the middle.
“Credit to Australia. They put us under pressure in there and we couldn’t respond.”
Australia’s David Warner was amongst those to question England’s mentality as they lost six wickets for nine runs in the first innings but Cook is certain his team can bounce back.
“This side has got plenty of character,” he said. “We’ve fought back in the past as well. Obviously it’s going to hurt us, this game.
“But there’s plenty of character in the side and we’re going to show that again.”