The hosts led 17-3 at the break through Danie Poolman's try, but they added four more scores in the second half to seal a bonus point, striking when Zebre - already depleted with 13 players on international duty - saw two men sent to the sin bin inside two minutes.
Connacht have now lost just one of the last four games and sit fifth in the GUINNESS PRO12 table.
And Lam was delighted with his team's showing in their biggest victory of the campaign.
"We couldn't win this game in the first ten or 20 minutes," said Lam, without seven first team regulars on international duty.
"We had to work and grind them down, try to control the ball and get some field position, so we could try and get our counter game going.
"When you consider we are missing Robbie Henshaw, Kieran Marmion and Rodney Ah You, it is great that we have that much quality to boost our squad."
The star attraction for the near 7,000 crowd was a first sight of full-back Mils Muliaina, capped 100 times by New Zealand.
The 34-year-old set up Poolman for the game's opening try, before a penalty try and scores from new team-mates John Cooney, Mick Kearney and Jack Carty wrapped up the victory.
And Muliaina, the first All Black to ever play for Connacht, was thrilled by his first taste of PRO12 rugby.
"The atmosphere was awesome," he said. "The supporters here are really passionate, especially in weather like this and to still have 5,000-odd crowd that turns up.
"We're pretty pleased and we're stoked about turning it on for them."
Click here for the best images from Connacht's win over Zebre Rugby
Follow us on Facebook, join the conversation on Twitter, sign up to our YouTube channel for extensive match highlights and sign up for our newsletter for regular updates on the GUINNESS PRO12
Guinness PRO12
Suite 208, Alexandra House,
The Sweepstakes
Ballsbridge, Dublin 4, Ireland