Despite being a goal down, Lambert felt his side's attacking threat was such that he had no concerns when they trailed at half-time.
"I thought we deserved the win, the passing was brilliant, and the enthusiasm and desire to press the game was excellent," he said.
The Canaries had trailed 1-0 at half-time thanks to Danny Graham's 23rd-minute opener but captain Grant Holt levelled two minutes into the second half as his header deflected in off Ashley Williams.
Anthony Pilkington put the visitors ahead just four minutes later as his strike hit Neil Taylor on its way in, before Holt grabbed what turned out to be the winner with a composed finish from a tight angle after being teed up by the excellent Elliott Bennett, taking his tally for the season to 11.
Holt blotted his copybook by giving away a late penalty for pulling back Williams, which Graham converted, but ultimately it did not prove costly as Norwich survived a fierce Swansea onslaught thanks to John Ruddy's brilliant save from Steven Caulker's point-blank header.
"Swansea scored against the run of play and maybe Andrew Surman should have made it 1-1 before half-time, but in the second half we were terrific," Lambert added.
"At half-time I said not to worry as we were well in the game, we gave as good as we got, the passing was excellent and the way we played the game was excellent.
"I was not too downhearted at all because I always think we are a threat. We were hanging towards the end when they got the penalty but we deserved the win."
The win moved Norwich up to eighth in the Barclays Premier League table, but Lambert insists his only target is survival, adding: "As long as we stay above 18th, that's it. If we stay in the league it will surpass anything me and my coaching team have done in League One or the Championship."
Source: PA
Source: PA