Osvaldo, a club record £12.8million signing from Roma, opened his account for his new club with a low 18-yard strike from the edge of the area two minutes into the second half.
Barely 90 seconds later, his strike-partner Lambert doubled Saints' advantage with a wonderful, curling 25-yard free-kick.
Visitors Palace, who have now lost three games in a row, struggled to create many openings on another testing afternoon for the newly-promoted side.
Their best opportunity came 28 minutes in when Marouane Chamakh went down in the area under a challenge by goalkeeper Artur Boruc, but the former Arsenal man was booked for diving by referee Martin Atkinson with replays later suggesting the official had got it right.
Saints came close to opening the scoring after just six minutes as Morgan Schneiderlin pounced on a loose ball in the box and looped a header towards goal.
It beat the onrushing Julian Speroni but was cleared off the line by Joel Ward just as it looked as though Mauricio Pochettino's side were about to take an early lead.
Palace were playing on the counter-attack and almost caught Saints out when Jimmy Kebe, making his first start for the Eagles, and Chamakh combined, with neither able to get a shot away on goal.
Lambert was the next Saints player to come close to breaking the deadlock after Adam Lallana rolled a ball to the England international which he curled just over the top of Speroni's goal.
The Palace goalkeeper was forced into action moments later as he got down low to punch Lambert's driven free-kick away. Osvaldo then miscued from Lallana's free-kick across the penalty area as the hosts looked to take control of the game.
But it was Palace's next attack that saw the most controversial moment of the first half as Chamakh gathered a Barry Bannan flick-on before driving in to the area and going down when through against goalkeeper Boruc.
Referee Atkinson deemed that the Morocco international had dived and was quick to produce a yellow card for the former Arsenal frontman.
With neither side really able to stamp their dominance on the game, the first half drifted towards a climax with only a long-range Victor Wanyama effort for the hosts producing a chance of note.
The second half started in much more dramatic fashion as Southampton rushed into a two-goal lead inside four minutes.
Firstly, Osvaldo opened his account for the club as he benefited from slack play from Palace captain Mile Jedinak.
The Australian was guilty of losing possession on the edge of his own penalty area and club-record signing Osvaldo latched onto the ball before wriggling free of Kagisho Dikgacoi and rifling past Speroni.
The home side did not have to wait long to double their lead as Osvaldo's strike partner Lambert also got in on the act.
Dikgacoi felled Osvaldo and the England international grabbed his second goal of the season as he bent in a perfect free-kick that Speroni could only throw a hand towards as it went in off the post.
Ian Holloway reacted by bringing on Cameron Jerome, Jerome Thomas and Kevin Phillips with the Eagles struggling to create anything in Southampton's final third.
Substitute Jay Rodriguez tucked away an effort late on but it was chalked off for offside before James Ward-Prowse shot high and wide as the hosts easily saw out the remainder of the game to win their third league game of the campaign.