Young striker Clayton volleyed home on the turn on 81 minutes when Ashley Westwood's corner was not cleared. And Murphy settled a scrappy match with a cool finish in stoppage time.
It is now two defeats in two matches for Paul Cook since taking over at Accrington last week.
Until Clayton's strike, the main talking point was referee Robert Madley's failure to award a penalty when Stanley defender Aristote Nsiala appeared to handball inside his own area on 27 minutes.
Greg Pearson had a shot blocked by Dean Winnard after he cut inside the Accrington box. Jamie Devitt tested Crewe goalkeeper Steve Phillips, however Madley enraged Alex further by booking Nick Powell for diving when he appeared to be tripped by Kevin McIntyre.
Accrington claimed handball against Pearson but once Clayton found the net, there was only one winner. Miller wrapped it up from the edge of the box after good work from Nick Powell.
