However, the hosts could not take any of several goalscoring chances, and enter the interval goalless with the Champions League-chasers.
He then stretched to divert a threatening Puncheon shot wide of the post as the Saints pressed in the opening stages.
Their dominance continued throughout the half as Lambert struck the woodwork with a beautifully-whipped 25-yard free kick.
The Saints leading scorer also saw a pair of headed efforts saved by Tim Howard, who later foiled Jos Hooveld from close range.
Gaston Ramirez also came close to breaking the deadlock with a low left-footed drive but Everton, who created little of their own, hung on to take parity into the break.