Richardson had claimed a win and a draw in his first two games since taking permanent charge of the Reds, and Stanley had led 2-1 early in the second half against Northampton.
New signing James Beattie was unable to make his debut after not joining the club in time to feature this weekend and the visitors took the lead in the 17th minute when Akinfenwa headed home a cross from Joe Widdowson.
Stanley levelled eight minutes before half time, though, as George Miller sent a swerving effort into the bottom corner from 25 yards.
Rommy Boco then put the Reds in front in the 57th minute, jinking past an opponent before lashing a left-footed strike beyond former Stanley loan keeper Lee Nicholls from outside the box.
But good work from Akinfenwa saw the striker lay the ball off for substitute Jake Robinson to make it 2-2 in the 69th minute.
Two minutes later the Cobbers were 3-2 ahead as Akinfenwa sent a diving header into the net from Emyr Huws’ cross.
A third Northampton goal in the space of seven minutes soon arrived as Akinfenwa completed his hat-trick, volleying home from a long throw.
Stanley: Ian Dunbavin; Dean Winnard, Rob Atkinson, Peter Murphy, Lee Molyneux; Luke Joyce, George Miller; Rommy Boco, Will Hatfield (Charlie Barnett 78), James Gray (Jack Sampson 90); Padraig Amond. Subs not used: Andrew Dawber, Toto Nsiala, Craig Lindfield, Michael Liddle, Danny Schofield.
Northampton: Lee Nicholls; Daniel East, Kelvin Langmead, Ben Tozer, Joe Widdowson; Lewis Wilson (Ishmel Demontagnac 63), Lewis Hornby, Chris Hackett, Emyr Huws; Clive Platt (Jake Robinson 63), Adebayo Akinfenwa (Henoc Mukendi 85). Subs not used: Shane Higgs, Anthony Charles, Ben Harding, David Moyo.
Referee: Andrew Madley.