The 32-year-old German has changed the team's fortunes since taking over as interim coach last month following Will Still's dismissal, guiding them to four wins in five matches.
"We go all in; there is no other way. I don’t shy away from saying that we want to play for promotion," Eckert said in a statement.
Southampton are 14th in the standings with 24 points from 18 matches, five points off the playoff spots. They host eight-placed Birmingham City on Saturday.