The away side found themselves 3-1 up just after half-time but the Cherries scored three second-half goals to get the victory in a game which saw Tommy Elphick and Joshua King come back from injury unscathed.
Under-21s striker Brandon Goodship was also introduced in the second-half to see his first action for over 18 months and he played a major part in the Cherries' winning goal.
The game was hugely affected by terrible conditions, but the Cherries dominated the early exchanges and enjoyed plenty of the ball. Despite the comfortable start, they found themselves 1-0 down against the run of play after 13 minutes - when Gregory Cundle tapped in from eight-yards.
The visitors doubled their lead when a swift counter attack lead to a strike from 20-yards sneaking past Ryan Allsop.
But the hosts were back in it ten minutes before the break when Glenn Murray went through and the Gills keeper parried his effort into the path of Iturbe, who took two touches to compose himself before smashing the ball in from ten-yards.
The Cherries dominated up until the second-half but found themselves 3-1 down just seconds after the restart.
Elliot List went through after Elphick slipped on the boggy surface and Allsop brought down the striker in the area. He stepped up to convert the penalty into the bottom right corner.
Elphick then completed his comeback from injury and was replaced three minutes before the hour for Callum Buckley.
On the hour, the Cherries were back in it again when Lewis Grabban crossed from the left and Iturbe had his second as he finished the striker's cross at the back post.
Seven minutes later, the equaliser came and it was nothing more than the home side deserved.
Murray rattled the crossbar with a volley before King reacted the quickest to nod in the rebound from close range. Straight after the goal, King and Murray were replaced by Goodship, back from a long injury lay-off, and Keelan O'Connell.
Goodship's return from injury was a successful one, playing a crucial role in helping complete the comeback. In the 71st minute he lashed a powerful effort towards goal from ten-yards that took a deflection off Iturbe and flew past the goalkeeper.
So it was a rather fortuitous way for the Argentine to complete his hat-trick, but there was no luck regarding the score as the Cherries were worthy winners in dreadful conditions.
Cherries: Allsop; Lee, Elphick (Buckley 57), Simpson (McCarthy 37), Cargill, Iturbe, MacDonald, O'Kane, Grabban (Cornick 64), King (Goodship 68), Murray (O'Connell 68).
Goals: Iturbe (35, 60, 71) King (67)