Christian Benteke saw a powerful penalty well saved by Tim Howard in the seventh minute and the American denied him again with a well-hit effort across goal.
Andreas Weimann somehow contrived to botch a straightforward one-on-one with the goalkeeper moments later and Everton went on to have great chances of their own, Ross Barkley hitting the crossbar with a deflected effort.
Lambert declined to attack his side for their early wastefulness, despite the result meaning they have taken just three points from a possible 15 on home turf.
The glut of early opportunities did not escape the Scot though.
"I think we have got to take our chances," he said.
"We should have been three up. But I can't be too harsh on the lads.
"The penalty was a terrific save, I've just seen it again, and the second chance Christian did the right thing in going across the keeper. He just made another great save.
"They're big moments if you don't score.
"Their goalkeeper had a really good game but you don't score when you're well on top and we got caught."
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