The game looked destined for a draw until left-back Luke Shaw burst into the Albion box with little over a minute remaining. A clumsy challenge from Youssouf Mulumbu sent the youngster tumbling and Lambert capped a dream week by smashing home the resulting spot-kick.
Both sides seemed up for the game and it was a frenetic encounter in the first period. However, the quality rarely matched the commitment and chances at either end were almost non-existent.
Southampton did have the ball in the net just after the half-hour but Adam Lallana's effort from Lambert's header across goal was judged offside by the referee's assistant, although the replays suggested Albion may have been fortunate.
The second half produced more goalscoring opportunities with West Brom just about having the better of the play. It was Southampton who came closest, however, as Jay Rodriguez clipped the top of the crossbar with a powerful right-foot drive from the edge of the area albeit the whistle had already gone.
When James Morrison saw an effort from point-blank range blocked, the game looked to set to end in stalemate but Shaw and Lambert had other ideas.
There appeared to be signs of early-season nerves in the opening stages of the game, with West Brom's number one Ben Foster making a hash of an attempted catch and Artur Boruc then giving away a corner by fumbling a Morrison shot.
Southampton did seem to be settling quicker, however, and Foster's palms were stung by a long-range James Ward-Prowse effort just before the quarter-hour mark.
At the other end Gareth McAuley nodded a Graham Dorrans corner well over, but Southampton were soon pressuring again.
Ward-Prowse's free-kick into the box fell to Lambert, who saw his shot on the turn well blocked by McAuley, and a scramble ensued, with Lallana also striking the ball against a Baggies shirt before the hosts got it away.
Albion attempted to get a grip on the game and, when Jonas Olsson nodded the ball to Mulumbu from a free-kick, the midfielder struck an effort wide off one of his team-mates.
Southampton then had the ball in the net as Lambert headed a delivery from Calum Chambers - making his first senior Saints start - back across the area for Lallana to poke in at the far post, but the assistant referee's flag was raised.
West Brom had their closest effort yet three minutes later, with Dorrans curling a free-kick around the wall and on to the roof of the net.
But Southampton remained the more likely-looking side to score and in the final moments of the first half Shaw saw his swerving shot from distance pushed behind by Foster.
The pattern remained the same as the second period got under way, with Liam Ridgewell nodding narrowly over his own bar before Lambert produced a header that was easy for Foster to gather.
Rodriguez was next up, firing against the bar, although play had already been stopped for a foul.
In the 56th minute Albion fashioned a decent-looking opportunity but Shane Long spurned it, opting to try a clever flick in the box that sent the ball wide.
Morrison then had a shot from the edge of the box blocked and moments later the same thing happened from closer range.
The ball went to the Scotland international in the danger zone from Jones and Morrison could not make the most of it, with Chambers getting himself in the way.
A couple of substitutions then saw West Brom take off one striker recruited over the summer in Nicolas Anelka and send another on in Matej Vydra, but it was the Saints who had the final say.
Mulumbu dragged down Shaw as he burst into the box and Lambert - as he so often does - kept his head to convert from the spot.
Jones almost found an equaliser in the last few seconds, but Boruc got down to deny him.