Jamie Ward saw his spot-kick saved for the second successive home game after a foul on Richard Keogh just before half-time.
Conor Sammon saw a shot strike a post just before the half-hour mark while visiting goalkeeper Scott Loach had to make a string of saves.
Derby could not make the breakthrough as they searched for a fourth success npower Championship victory but it was Ipswich who nicked all three points right at the end with Edwards' long-range strike.
The Tractor Boys kept a fifth straight clean sheet and recorded an eighth shut-out in their last 10 league matches as they extended their unbeaten run to five games.
Jamie Ward and Craig Bryson were both denied by Loach before Sammon struck a post after cutting in from the left in the 29th minute.
Derby were on top but at the other end David McGoldrick's rising drive was held by Frank Fielding, while Daryl Murphy just failed to connect with Edwards' low cross.
In the 44th minute, Richard Keogh was adjudged to have been pushed by Tommy Smith following a corner.
Smith, who had just been booked, escaped further punishment and Ward's low penalty was kept out by the legs of Loach.
Derby had the ball in the net just after the hour mark through Chris Martin from Sammon's low cross, but the on-loan Norwich striker was flagged offside.
Jake Buxton was next to go close for the Rams but his far-post header from Paul Coutts' corner went just wide.
The hosts paid the ultimate price for failing to take their chances when Edwards let fly from 25 yards out and the ball flew past Fielding and into the bottom corner of the net.