The Spireites, bouncing back from a four-goal mauling at Oldham in midweek, twice punished Posh from set-piece situations but then had to dig deep and hang on in the final quarter of an hour after Curtis Woodhouse pulled one back.
Midfield operator Sammy Clingan was the man who unlocked the home defence, first providing a 24th minute free-kick that Downes headed in after goalkeeper Mark Tyler could only push the ball against he back of Chesterfield striker Caleb Folan.
Clingan struck again nine minutes after the break with Skipper Evatt forcing the ball home in a goalmouth scramble as Posh failed to clear their lines once more.
Although Posh opened brightly with recent recruit Wayne Purser forcing Carl Muggleton to spread himself to save well, they lacked confidence going forward and, despite forcing a succession of corners, the experienced Muggleton was rarely troubled.
Chesterfield might have added to their lead just after the break with Tyler making an impressive save from Folan and Posh boss Barry Fry responded in typical style using all three substitutes, among them 30-year-old former Spurs star Darren Caskey who was cautioned for a foul barely seven minutes into his return to league football.
Caskey's play making from midfield did make a difference and as Posh went on the offensive, captain Woodhouse ensured a grandstand finish converting David Farrell's swirling cross with a fine header.
Posh kept up the pressure but Chesterfield defended in depth and managed to hang on to their odd-goal advantage through a full five minutes of added time.