The visitors were dominant at the Millennium Stadium but were unable to turn possession into points.
England manager Johnson told Sky Sports 2: "It was a bit of a mess out there. No excuses - we have to sort it out.
"We lost a little bit of cohesion. We had a stranglehold on the game and we let it slip away. It's really disappointing.
"It's one of the strangest games I have seen.
"We will sort it out and get better."
Sam Warburton admitted Wales had to hang on "for our lives" in the first 40 minutes.
Man-of-the-match Warburton said: "England were fantastic in the first half. We hung on for our lives. We got the result, that's the main thing."
Warburton said it was a "massive" result in terms of the World Cup campaign.
"A win was vital," he said. "We have to get in the winning habit."
England captain Mike Tindall said he was disappointed with the result but would "take it on the chin".
"We ended up getting in to an arm wrestle which is not what we wanted today," he said.
"We had so much ball, so much possession and we didn't come away with anything. We have to go back and review things properly. We have to take it on the chin.
"We came here to work on things but did not get as many points as we should have."
Wales coach Warren Gatland said: "In terms of commitment I cannot fault the players - they were absolutely tremendous.
"I thought as the game went on we looked very strong.
"Today was not about pretty rugby, it was about showing character."
Gatland said Wales should not be underestimated, adding: "That's the message I want to get out to the public."
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