The England international scored the second goal in the 2-0 win over Crystal Palace - a result which lifted Mauricio Pochettino's side up the giddy heights of fourth
"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."
Club-record signing Osvaldo finally opened his account for the club against Palace and Lambert believes it will do the former Roma man a power of good.
"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."