Michael Morrison opened the scoring for the Blues, before Lee Tomlin netted a penalty just before half-time. City striker James Vaughan missed a second-half penalty before substitute Glenn Murray's late winner ensured the Cherries were victorious against the Championship side.
Manager Eddie Howe made 11 changes for the cup tie and introduced academy prospects Jordan Lee and Matt Butcher for their first professional starts, as well as giving youngster Baily Cargill the captain's armband.
The pair had to do some defending as the home side pushed forward in the early stages, with Andrew Shinnie firing an effort over in the area before he lashed an effort across Adam Federici's goal.
But on 22 minutes the Cherries were inches away from going ahead, when Marc Pugh beat his man on the right hand side, dinked a ball in for Yann Kermorgant who rose above his man to nod the ball onto the bar from eight-yards.
The visitors were in the ascendency as Tokelo Rantie then hobbled off to be replaced by Matt Ritchie, but after plenty of attacking pressure, the Midlands side took the lead five minutes before the break.
David Cotterill whipped in a free-kick from the right-hand side and Morrison powered in a header which found the bottom right corner from eight-yards.
Howe's men wasted no time in finding the equaliser though before half-time as a minute before the break, Eunan O'Kane was bundled over in the box and referee Michael Jones pointed to the spot with no hesitation.
Tomlin stepped up to fire a low spot-kick down the middle to give the Cherries the equaliser and parity at half-time.
The second-half became a scrappy affair as the conditions worsened and the Cherries were given a massive let off in the 58th minute.
O'Kane, who won the penalty at the same end, gave one away as he went into a challenge with Maikel Kieftenbeld and the foul was given.
But Vaughan stepped up and blazed his side-footed penalty well over the bar to let the Cherries off the hook.
The Blues then had yet another chance to take the lead as substitute Jacques Maghoma robbed the ball off Cargill and got free on the left-hand side of the box, but his curling effort struck the right hand post with Adam Federici beaten.
But Murray came on for Kermorgant shortly after and his striker's instincts gave the Cherries the lead with just five minutes to go.
O'Kane headed a cross from the right which keeper Adam Legzdins did excellently to palm away but Murray was first to sniff out the rebound and pop the ball in from close range at the back post.
Ritchie struck the post late on with a header and the Cherries survived a lot of late pressure but in front of a massive away following, Howe's men were victorious to secure safe passage into the fourth round for the third year running.
Cherries: Federici, Lee (Cook 83), Cargill, Distin, Butcher, Tomlin, MacDonald, O'Kane, Pugh, Rantie (Ritchie 39), Kermorgant (Murray 78).
Unused subs: Allsop, Ward, Green, Wakefield.
Birmingham City: Legzdins, Eardley (Lowry 45), Morrison, Spector, Grounds, Kieftenbeld, Davis, Cotterill (Maghoma 45), Shinnie, Soloman-Otabor (Toral 68), Vaughan.
Unused subs: Kuszczak, Gleeson, Donaldson, Brock-Madsen.
Goals: Morrison (40), Tomlin (44 pen), Murray (85)
Attendance: 13,140 (2,354 away)
Referee: Michael Jones
Yellow cards: N/A