The home side dominated the opening 45 minutes and could have been out of sight by the interval, but a mixture of poor finishing and good goalkeeping from Erwin Mulder ensured that the spoils were shared.
Birmingham are still without a win in four matches this season, taking just two points from a possible nine in the league, while Garry Monk's former employers join fellow unbeaten side Middlesbrough on seven points at the top of the early standings.
Swansea should have been full of confidence following wins over Sheffield United and Preston North End in their first two outings under Graham Potter, but Birmingham completely dominated the first-half chances.
Cardiff City loanee Bogle then forced Mulder into a routine stop on the Dutchman's first Swansea start, before slotting one wide, while Jota volleyed over the crossbar soon after.
Birmingham were the Championship's joint-lowest scorers last season, and it was becoming clear to see why as Jacques Maghoma fired a shot wide and Lukas Jutkiewicz missed two glorious openings, the second from a one-on-one position.
Mulder then kept out Jota and Dean in quick succession, but only after a big scare when the hosts finally managed to get the ball over the line.
Jutkiewicz's header was looped into the air by Mulder and dropped under the crossbar, only for the officials to rule the goal out as Michael Morrison made illegal contact with the keeper and was also standing in an offside position.
Birmingham would rue their missed chances, as the second half played out with minimal opportunities for either side.
Bersant Celina was wayward with his attempt from range and Jota had a volley blocked from a similar position.
Birmingham did step things up in the final 10 minutes, with substitute Che Adams smashing the ball over the bar from a tight angle and Jutkiewicz turning on the edge of the box but meeting his match in Mulder.
A frustrating night for Monk's men would finish goalless, leaving them in the lower reaches of the second tier.
BIRMINGHAM CITY (4-4-2): Camp; Colin, Morrison, Dean, Pedersen; Maghoma, G.Gardner (Lakin 85'), Kieftenbeld, Jota; Jutkiewicz, Bogle (Adams 79')
SWANSEA CITY (4-2-3-1): Mulder; Roberts, Van der Hoorn, Rodon, Olsson; Fulton, Carroll (James 64'); Asoro (Grimes 77'), Celina, McKay (Fer 64'); McBurnie