The striker had already been booked when he was given his marching orders by referee Iain Williamson for tripping winger Alefe Santos as he threatened to break down the right flank.
It was a bad day for Doyle, who also missed his side's best chance at the start of the second half, chipping over the crossbar when Armand Gnanduillet's pass put him through down the middle.
The visitors looked more likely to break the deadlock and Rovers goalkeeper Steve Mildenhall did well to block a 64th-minute shot from Gary Roberts with an outstretched leg.
Rovers saved their best until injury time when a fierce 20-yard drive from Ollie Clarke was beaten away for a corner by Spireites 'keeper Tommy Lee.
The young home side battled hard throughout but Chesterfield will be disappointed at not clinching all three points as League Two's early pace-setters went six games without a win.