Dundee have taken only one point from their last three games and were overtaken at the summit by Falkirk.
A 1-1 home draw on Saturday with Alloa, who were led by Brown's predecessor Barry Smith for the first time, proved the former Rangers player's final match in charge.
A statement on the club's official website read: "Dundee FC and manager John Brown have today announced they are to part company by mutual agreement. The club agreed with John's expressed view that after recent results, this is the best way forward."
Dundee are level on 40 points with Falkirk but the club are desperate to go up this season ahead of the likely arrival of Rangers and Hearts in the second tier.
And they could have a replacement in the dugout when they take on title rivals Hamilton, who sit a point behind in third place, at Dens Park on Saturday.
The statement continued: "John stated that he has loved his time at Dundee FC as a player and as a manager. He still passionately wants Dundee FC to win the league this season and would share in our pride if we go on to do so.
"The club expects to move quickly to appoint a new manager, to continue the good work by the management and players to date and to and get us over the line as champions in May.
"We urge the fans to continue to support the club in the superb numbers they have done so far this season."
Brown was a surprise choice to take over from Smith on February 23 last year, initially on an interim basis, when the club were 15 points adrift at the foot of the table.
But they lost only once in his first nine Scottish Premier League games in charge to push St Mirren until the third last game of the season, a controversial penalty against Aberdeen ultimately sending them down.
Dundee recovered from an indifferent start this season to overtake Hamilton at the top of the table but the Dundee board have decided they want a fresh approach to edge them through a three-way title battle.
The club statement added: "The board of directors would like to put on record their thanks not only for his valiant efforts in attempting to keep us in the Premier League last season when no one gave us a chance, but also for his hard work and diligence in achieving the current points total which sees the club joint top of the league with 14 games of the season to go.
"His work with the youth teams has also seen a dramatic turnaround in the club's fortunes with the reserves team also second place and a number of our youths in the running for national honours."
Dundee have twice sacked managers when top of the First Division - Jocky Scott on March 20, 2010 and John McCormack on February 9, 1998.