Adam Lallana added Southampton's second and there might have been several more as Tranmere wilted after a promising start.
Rovers held out for 43 minutes, that was until Danny Butterfield crossed from the right and Lambert headed in from six yards.
Southampton added a second in the 57th minute when Brazilian midfielder Guly Do Prado set up an easy chance for Lallana to score with a left-footed shot.
Lambert was denied another late on by a superb save from Peter Gulasci while Richard Chaplow shot wide when clean through.
Lambert also hit the bar in the first half as Tranmere struggled to make any impression.
The visitors failed to muster a single shot on target apart from an effort by Lucas Akins in the 18th minute, his shot held by Kelvin Davis.
Southampton manager Nigel Adkins took off Guly, who received a standing ovation for his contribution, and sent on Lee Barnard who should have scored at the end but shot straight at Gulasci when played clear by Dan Harding.