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 successive 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.
Both Ward and Craig Bryson were named in the Derby starting line-up.
Leading scorer Ward returned to the team after missing Monday's win over Leeds with a tight hamstring. The 12-goal forward replaced Ben Davies.
Midfielder Bryson was fit despite hobbling off during the first half at Elland Road, so the Rams were otherwise unchanged.
Gareth Roberts was back in the squad after a calf injury but had to settle for a place on the bench as on-loan midfielder Craig Forsyth continued to deputise at left-back.
Ipswich boss Mick McCarthy made five changes following the goalless draw at Millwall on Monday - reverting back to the same starting XI that beat Leeds 3-0 last Saturday.
Skipper Edwards, Andy Drury, Jay Tabb, David McGoldrick and Daryl Murphy were recalled.
Jay Emmanuel-Thomas, Frank Nouble, Michael Chopra and Guirane N'Daw dropped to the bench, while Lee Martin missed out completely having been ill this week.
Ward and 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, 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.
- Sports & Recreation
- Jamie Ward
- Scott Loach
- Conor Sammon
- Craig Bryson