The Tykes, who had the second-lowest average attendance of 10,207 in the Championship last season, defied the odds on the final day to survive another year in the second tier.
In an open letter to all Barnsley fans on the club's official website Flitcroft said: "You voted with your voices to put me in this post as your manager.
"You will never know what that means to me and how proud my family was of what we all achieved together as a club.
"You have welcomed me and my staff into the Barnsley family. Thank you, I will give you my all."
Barnsley drew 2-2 at Huddersfield on the final day to escape relegation having looked odds-on for the drop when Flitcroft replaced former boss Keith Hill at the turn of the year.
The 39-year-old added: "We will have to overachieve next season. Overachieving is about everyone working as one.
"Can we, as a town, overachieve before the season and get 10,000 fans to commit to a season-ticket?
"Anything is possible if you dare to dream, dare to believe and dare to do.
"Together we can dream, together we can do, together we can believe and together we will achieve."