The Saints gained promotion to the English top flight after finishing second in the Championship in the last campaign, but are one of the candidates to go down.
Adkins admits that it will not be easy for Southampton to retain their Premier League status, but insists that his players have what it takes to stay up.
"Everybody should be looking forward to it, enjoying it," Adkins told The Independent. "People questioned [striker] Rickie Lambert before last season, his first in the Championship, and look how well he did. You have to have belief, go and grasp the opportunity that we have worked very hard to get.
"You know you are going to play against teams and players who are better, so you have to be as smart as you can be to deal with that, try to play the game on your terms and not theirs.
"Not everyone was expecting us to do as well as we did in the Championship, but we were in the top two the whole season - on merit. Yes, we are going to the next level, but there is a stability here that should give us strength. You can't underestimate team spirit."
Southampton will begin their Premier League campaign with an away game against champions Manchester City on Sunday, August 19.