Murphy was on the end of good work from Paul Taylor and Frank Nouble to tap into an open goal on 38 minutes as Ipswich won their first match in four games.
A woeful first half an hour came into life thanks to Murphy's opener and Ipswich deserved their three points after an impressive second-half performance.
While the home side enjoyed the lion's share of possession early on, it was Birmingham who had the better chances.
The visitors had the best chance in the opening 20 minutes when Lee Nowak showed a great touch to bring down Jonathan Spector's long ball, but his strike flew just wide of the post.
On the half-hour mark, Ipswich finally mustered their first attempt at the Birmingham goal.
Aaron Cresswell's teasing free-kick was met by Christophe Berra but he was unable to direct his header on target from a good position.
In a poor first half, Nouble went close to opening the scoring eight minutes before the interval when he beat Spector but Darren Randolph made a comfortable save at his near post.
Less than a minute later, Ipswich took an unexpected lead after Taylor and Nouble combined and the latter's low cross was tapped in at the far post by Murphy.
The goal galvanised the previously-underpar hosts and as the half drew to a close, Town's frontmen again combined well but Randolph was equal to Nouble's dangerous low cross.
Ipswich came out for the second half looking to put the game to bed and eight minutes after the restart, livewire Taylor brought out the best in Randolph who was equal to a curling effort destined for the far corner.
At the other end, Birmingham sub Jordon Ibe gained himself some space but his shot was comfortably held by Dean Gerken on 56 minutes.
It was the home side who looked the more threatening with Taylor again missing the far post, this time by a whisker, with another devilish right-footed effort.
Birmingham should have been level after 63 minutes in a similar chance to Murphy's goal. Chris Burke somehow beat Tommy Smith to the ball but his low cross was turned over by Nowak from close range.
The second half was a much brighter spectacle and Luke Chambers came within a whisker of doubling Town's lead but his delightful left-footed curler just missed the far post.
Sub Paul Anderson then went even closer, latching on to Spector's dismal back pass but Randolph pushed aside his effort after the winger looked to have rounded him.