Wednesday, looking for a quick fix after a 2-0 defeat against South Yorkshire rivals Doncaster in midweek, got off to an explosive start.
Marcus Tudgay found Luke Varney whose crisp volley from the edge of the area was pushed from under the bar by Murphy.
The visitors should have taken the lead after the hosts failed to cut out Edwards' cross as centre-half Damien Delaney headed wide from close range.
Roy Keane's side broke the deadlock in the 15th minute, Edwards was given too much time and space as he lashed ball into the far corner from 20 yards.
Jack Colback fired straight at Owls keeper Lee Grant and midfielder Grant Leadbitter dragged the ball narrowly wide before the woodwork saved Ipswich. Tudgay created space and drilled an effort towards goal but the bar came to the Tractor Boys' rescue.
Michael Gray was the latest to be thwarted by Murphy as he stopped an effort that was destined for the net.
A mistake by winger Etienne Esajas let in Edwards on 58 minutes, but the livewire striker had his shot saved by Grant.
Jon Walters headed in at the far post from a Leadbitter corner but the goal was disallowed for a push before Varney almost grabbed an equaliser, pulling his overhead kick narrowly wide in the dying minutes as Ipswich held on for a vital victory.