The Irishman leaped highest to head home a Paul Anderson delivery after Matt Ritchie had previously regained the home side’s advantage.
Bournemouth had already been pegged back once before after Yann Kermorgant saw his spectacular opener cancelled out by Teddy Bishop.
It took less than two minutes for Eddie Howe’s side to take grab the lead at Dean Court as Kermorgant gave the hosts the advantage in style.
The former Charlton striker met a Simon Francis cross perfectly and made no mistake as he acrobatically smashed the ball past Bartosz Bialkowski from outside the box.
Ritchie had almost given the Cherries’ the lead moments earlier as he crashed an effort onto the Tractor Boys’ crossbar with a left-footed effort.
Bournemouth had the chance to double their lead with just nine minutes gone as Marc Pugh went one-on-one with Bialkowski in the Ipswich goal.
But Town’s stopper, dubbed Bart by the Ipswich players, closed the gap well to save the shot before getting up again to block Ritchie’s second bite of the cherry.
It took 20 minutes for Mick McCarthy’s side to get involved in the game and they fired a warning shot via Murphy.
Murphy leapt highest from a Town corner to power a header past Artur Boruc, and the Southampton loanee was thankful to see it crash off the bar and away to safety.
The second-half at Dean Court started as frenetically as the first, only this time it was Ipswich that were fastest out the traps.
The hugely impressive Tyrone Mings fed substitute Bishop who slotted the ball into the bottom right-hand corner of Boruc’s goal to equalise.
Bournemouth however were in no mood to drop their first home points in five games and responded almost instantly.
The impressive Ritchie again involved as he tapped home a Callum Wilson cross after the former Coventry striker had taunted the Ipswich backline with his pace.
But Ipswich were showing an air of confidence that not many sides’ have possessed at Dean Court this season.
And the Tractor boys were handed a free kick just outside the box when left-back Mings was wrestled to the floor by Francis.
Murphy yet again beat every Bournemouth defender to the delivery and headed the ball powerfully past Boruc.
Injury to substitute Tokelo Rantie meant that there was 12 minutes of added time to be played at Dean Court.
The hosts pushed for the winner and midfielder Harry Artur had the chance to snatch all three points with minutes left be fired tamely at Bialkowski.