The Saints beat Crystal Palace 2-0 on Saturday, with Lambert on target alongside Dani Osvaldo, which has them sitting in fifth spot after six matches.
Lambert knows it is a very positive beginning by Southampton but says they have not done near enough in the early stages 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 in the top five) 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."