The ex-Southport manager, 37, joined Brabin's staff at Prenton Park in July.
He will be assisted by first-team coach Shaun Garnett and head of academy and recruitment Lee Jones.
It is the second time that Carden has succeeded Brabin, with his promotion to Southport boss coming after Brabin joined Everton's coaching staff.
Carden's first match in charge will be against Woking on Saturday, with Rovers fifth in the table after 11 matches.
Assistant manager Paul Carden has taken charge of National League side Tranmere Rovers on an interim basis following the sacking of Gary Brabin on Sunday.