Gus Poyet's side continued their excellent form in knockout competitions as the League Cup finalists reached the last eight, but only after Lambert somehow contrived to clear the crossbar from two yards in the closing stages.
Gardner's stunning strike five minutes after the break proved decisive, the midfielder turning away from Victor Wanyama before firing home from outside the penalty area.
Sunderland had several chances to make it more comfortable but none as good as the one presented to England man Lambert, whose miss let the hosts off the hook.
Poyet, whose men remain embroiled in relegation trouble, promised to ring the changes ahead of a busy schedule for his side and he duly obliged, making nine alterations as forward Ignacio Scocco was handed his debut.
Mauricio Pochettino has no such problems, with his charges eighth in the table, but he made six changes as both sides struggled to find their rhythm in a frenetic opening.
The visitors had the first opportunity, Adam Lallana finding Lambert with a cross before the striker nodded down for James Ward-Prowse, whose effort was well saved by Oscar Ustari.
Poyet was angered by the officials' decision not to award a penalty when Fabio Borini went down under the challenge of defender Luke Shaw after 12 minutes.
Emanuele Giaccherini then squandered a promising opportunity when he failed to get a shot away after dribbling his way into the Southampton area.
Lambert was denied an excellent chance to put his side ahead eight minutes before the break when Santiago Vergini executed a well-timed tackle in the box.
Gardner sparked the game into life with 50 minutes gone, putting his side ahead with a long-range strike that crashed in off the underside of Kelvin Davis' bar.
It was Gardner's third goal of the season and the visitors struggled to mount a response, with Pochettino opting to introduce Jay Rodriguez and Morgan Schneiderlin just after the hour mark.
Connor Wickham should have made it two 17 minutes from time but he failed to connect with a pass when through on goal, while Lallana could only put Nathaniel Clyne's cross wide from close range.
Pochettino must have realised it was not to be Southampton's day when Lambert spooned Clyne's low cross over the bar with the goal at his mercy as Sunderland sent the visitors crashing out of a second cup competition this season following their 2-1 League Cup win in November.
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