Gareth McAuley's strike put Roy Keane's side ahead in the fourth minute before Carlos Edwards added a second with nine minutes to go.
Rob Hulse pulled one back to give the home faithful hope, but Stephen Pearson's late dismissal ended the match as a contest before Connor Wickham wrapped things up.
Nigel Clough's side were unbeaten in five matches going into the game, but it was the visitors who started the brighter with youngster Wickham forcing an early save from Stephen Bywater.
The Rams did not take the warning seriously, though, and Ipswich took the lead just a minute later.
The hosts could only clear Grant Leadbitter's left-wing corner to McAuley, who grabbed his fifth goal of the season with a deflected left-foot strike.
County responded well, though, and nearly got a leveller in the 12th minute but Brian Murphy denied Steve Davies.
The Tractor Boys continued to cause the Rams problems with Wickham looking especially threatening.
And they should have doubled their lead in the 32nd minute only for Daryl Murphy to hit his shot straight at Bywater from six yards.
Ipswich continued to create chances and again could have grabbed a second but Wickham's effort went inches wide after McAuley found him in space.
Derby desperately tried to stop the visitors' relentless pressure but continued to struggle as Leadbitter's excellent 25-yard strike clipped the outside of the post.
Clough's side rallied after that chance, but Brian Murphy produced an excellent save to deny Shaun Barker.
The Rams picked up where they left off after the interval with an equaliser looking increasingly likely as the game wore on.
Sunderland loanee Jack Colback did his best to restore parity midway through the half but Murphy was equal to his 25-yard effort.
Substitute Hulse then spurned a great opportunity for the home side, heading straight at Murphy.
And the hosts were made to pay for their lack of composure in front of goal as Edwards got on the end of Wickham's excellent ball to put Ipswich two up.
Undeterred, Derby pushed for a goal and got it through Hulse, who rose highest to nod in Robbie Savage's cross.
The goal seemed to turn the game sour, though, as first Clough was sent to the stands after an argument with the referee before midfielder Pearson received a straight red card for foul and abusive language.
And Clough's misery was complete in injury time as Wickham raced through and fired home from 12 yards.