The hosts fell behind early when Christophe Berra headed home Aaron Cresswell's cross.
Zak Whitbread, making his debut since joining on loan from Leicester City, headed home to reduce the arrears.
However, Cresswell continued his great opening to the game by getting Ipswich's third goal before a quarter-of-an-hour had been played.
Things got worse for the Rams as Cresswell was again involved, turning provider for Murphy to net his second and Ipswich appeared to be out of sight before half time.
Darren Wassall, who temporarily took charge of the first team for tonight's game before Steve McLaren assumes the role, made a double substitution with the introduction of John Eustace and Mason Bennett.
The changes appeared to have an effect as Bennett played in Craig Bryson, who scored 97 seconds after the restart.
Derby dominated possession and teenager Will Hughes helped to pull another goal back as he found Jamie Ward who scored to give Ipswich manager Mick McCarthy plenty to worry about for the final 29 minutes.
Indeed, the hosts appeared the more likely to score as they pressed for an equaliser and they got it with two minutes of the 90 remaining as Ipswich failed to clear a corner and Bryson firing the ball home to the delight of the Pride Park crowd.
In the end, both sides took home a point, although Mick McCarthy may feel his side threw away a comfortable lead after their first-half showing.