The England international scored the hosts' second goal in a routine 2-0 win over Crystal Palace at St Mary's, just two minutes after strike partner Dani Osvaldo grabbed his first Southampton goal.
The win lifted Mauricio Pochettino's side into the Champions League qualification places, leapfrogging the likes of Manchester City, Liverpool and Everton in the process.
But 31-year-old Lambert knows Saints need to maintain their form if they are to mix it with the big boys at the business end of the campaign.
"A good start is important to any team in any season, but just because we've done well doesn't necessarily mean we can lay off a little bit," he said.
"We'll be trying to improve because we know the teams around us in the league are going to pick up points so it's going to get harder to stay where we are, but we're confident we can do it.
"It's very nice [to be fourth], so let's try to stay there now. It's early doors, so we've got to take our start into the rest of the season and push on so we can stay there."
The goals came in the opening four minutes of the second half as club-record signing Osvaldo finally opened his account for the club and Lambert then curled home a superb free-kick.
Lambert believes it will do the former Roma forward the world of good to break his duck.
"Dani did terrifically for his goal," he said.
"He's hit it, got it on target and it's gone in the back of the net. That's what happens when you take your chances, and that'll obviously give him the confidence he needs - and it gave the team confidence to go on and get another goal."
Palace boss Ian Holloway also believes confidence is required amongst his new-look squad as they look to add to the three points they have on the board from their opening six league games.
"I've had to pick 25 [players] and in that 25 there are 12 new ones," he said.
"You haven't got to be a genius to work out that we have got to learn about each other and work in the toughest division in the world. We have got to try and get some points.
"Next week we have got Liverpool away. This team [Southampton] have just beaten Liverpool away and I saw enough from my team in that first half that, if we get an opportunity and score, then confidence will come and we will look a bit brighter.
"The group has got to stick together and what you will find from me is that I won't be blaming my group, I will be taking the blame myself."
Holloway also admitted he would sit down with Marouane Chamakh to talk over the Morocco international's booking for diving in the first half.
The former Arsenal man was through on goal but went down as he looked to round Southampton goalkeeper Artur Boruc, with referee Martin Atkinson booking Chamakh instead of awarding a penalty.
Holloway felt there was contact but believes Chamakh should have taken the chance, although the 50-year-old confirmed any disciplinary action taken against the forward will remain private.