The South Coast club, who travel to Chelsea on Sunday, have taken 22 points from their first 12 Premier League fixtures this season.
Such fine form sees them sit above defending champions Manchester United in the table.
"I'm not surprised," Lovren told reporters. "I was believing from the first day when I came here to Southampton that we can do an amazing job and be a part of the other big clubs.
"We have the confidence now and we need to work harder and that is it, after that I think we can be at the top.
"I think for everybody it is an ambition to play in the Europa League or Champions League, that is our dream. If I can choose of course I will take the Champions League, we need to just believe in it and we can make it."
Lovren, who moved to St Mary's from Lyon for a transfer fee of £8.5m in June, has started all but one of Southampton's league games this season.