It was a battle of the boots as Craig Willis was outscored by Jack Carty by three penalties to one in the first-half.
Richard Mayhew finally broke through for Newcastle, but after Carty struck thrice more to close the match out, winger Danie Poolman put the gloss on the victory with Connacht's first try late on.
And after a first Challenge Cup win over an English side since 2010, Lam was full of praise for his determined squad.
"I am very, very proud of the guys," he said. "It has just been horrendous with the injuries we've had.
"Then we lost Tiernan O'Halloran. We had Matt Healy, who has never played full back, operating at this level.
"We had six academy guys in there too and they all acquitted themselves well.
"The thing I love about this team and these guys is that they are all hungry, they all want to be a part of it.
"We had to grind it out, it was a tough game. The conditions, it was pretty soft underfoot. We put some pressure on them and didn't quite finish things off."
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