Piggot was unveiled on Tuesday night following Anslow's exit in the wake of their 7-0 hammering at home to Hastings United on Saturday.
Anslow and his backroom team - including goalkeeping coach George Howard - left the Maidstone Road Sports Ground on Sunday following the heavy defeat and a run of inconsistent results that had left the Chats third from bottom.
Director of football Steve Binks had been expected to take charge before the appointment of Piggot who's also worked for Ryman South rivals Faversham Town and Lordswood as well as managing Gillingham-based Southern Counties East League outfit Hollands & Blair.
A club spokesman said: "The board of Chatham Town Football Club are delighted to announce that former Hollands & Blair manager Paul Piggott has accepted the role as new first team manager following the departure of Darren Anslow.
"Paul was instrumental in Gillingham club Holland and Blair’s recent successes and has previously worked at other Kent sides Faversham and Lordswood and his appointment was ratified at a committee meeting this evening, Tuesday 22 November.
"The chairman and committee of Chatham Town would like to welcome Paul to the club and wish him a happy and successful tenure."
Anslow and his team were also thanked for their efforts since taking over from Tony Beckingham in September by Chatham Town chairman Jeff Talbot.
He said: "We thank Darren and his backroom team for their efforts to turn things round since taking over a few months back. I would like to wish them well going forward."
Piggot takes the helm for the first time on Saturday at Whyteleafe with his new charges looking to secure only a second away win of the season.
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