The club have not won any of their three matches since the Argentinian replaced Nigel Adkins, who was sacked last month.
However, the former Bristol Rovers and Rochdale hitman is confident Saints will stay up and has backed chairman Nicola Cortese to provide the funds for the new man in the summer to help the club challenge the big boys.
The 30-year-old told The Sun: "The new manager has been great and you can see since he has come to the club he wants to make things happen.
"The chairman is very ambitious and Southampton is going to be a very exciting place to be over the next few years.
"It is not unrealistic to try and challenge for the top four over the next three years. That is what we have to aim for.
"Three years ago a lot of people would have laughed in our faces if we had said we'd be in the Premier League.
"We have got to keep that momentum going and keep moving forward. I am very confident we will stay up."
Lambert is also hopeful of joining club legends Mick Channon and Matt Le Tissier by scoring 100 goals for the club and he is aiming to do it against Manchester City.
The striker, who has bagged 11 goals this season making him the highest-scoring Englishman in the Premier League, said: "Scoring my 100th goal against a world-class team like City would be special but only if we get a good result.
"I try not to think about the record too much because the most important thing is the team get three points to move away from relegation danger."