First-half goals from Joe Bryan and Curtis Nelson gave the visitors a two-goal half-time cushion and, although Guy Branston put through his own net 12 minutes after the break, Plymouth held on for three precious points.
Argyle started well and had already created two chances before Bryan put them in front with a brilliant curling shot from the left side of the box in the 21st minute.
The visitors increased their lead five minutes later when Chesterfield failed to get the ball away and Nelson pounced to fire into the roof of the net from close range.
The hosts had offered little in the first half but they were handed a lifeline in the 57th minute when Branston turned a low cross into his own net.
They had plenty of possession but could not find a way through a stubborn defence and Andres Gurrieri missed a good chance to score a third late on.