Ipswich went ahead with their first major chance in the ninth minute, when Murphy was left free at the back post to tuck in from Christophe Berra's headed flick on.
Murphy came close to adding a second soon after as he was picked out by Tommy Smith, but the striker fired wide from a tight angle on the right side of the box.
The Town striker appeared in great form in the first half, and he soon returned again to force a good stop from David Button down low following a long-range strike.
Brentford almost pulled level through Pritchard when he fired towards the top corner, only to draw an equally good save from Town keeper Dean Gerken.
The visitors found an equaliser just a few minutes later when Stuart Dallas had his shot parried, leaving Douglas to head home from the rebound.