Sparrow struck early on after a Gary Hooper counter-attack before Hayes increased their comfort margin with a sharp volley nine minutes after the break that put enough distance between themselves and John Sheridan's men to make Darren Currie's 72nd minute penalty nothing more than a consolation despite some late pressure.
In front of a meagre crowd of 2501, the home side took the lead with 12 minutes gone through Sparrow.
Chesterfield had Paul Harsley off the pitch injured when the goal came in the form of Hooper breaking away down the right to square for the unmarked winger, just seconds after Donal McDermott's corner had been headed off the line by Marcus Williams.
Chesterfield, also opening day losers at Torquay, had Lee to thank for keeping the half-time score to one, with his stunning one-handed save diverting Michael O'Connor's drive to safety, while the Spireites also had some opportunities before the whistle, the pick of which saw Martin Gritton flick a header inches wide from close range.
O'Connor had a 35-yard free-kick disallowed two minutes after the restart, with the linesman catching Hayes offside as the Northern Ireland international's delivery caught everyone off guard to float into the net.
The clinching second goal was soon to follow, though, with Hayes volleying home from a tight angle after Lee could only parry another fizzing free-kick from O'Connor into his path. Hayes, who hit 19 goals last season, was then unfortunate to see a curling effort from 25 yards cannon back off the crossbar.
The visitors got one back when substitute Currie sent Josh Lillis the wrong way from the spot after Rob Jones had tripped Drew Talbot in the area.