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Aaron Cresswell tried to set up the visitors for an early goal, but a foul inside the box by the attacking team prevented Ipswich from getting the chance.
However, Cresswell was able to deliver the ball back into the box later in the first half, dropping to Christophe Berra, whose powerful finish from close range gave the Tractor Boys the lead.
Cresswell continued to cause problems for the Birmingham defender, blasting a shot just over before sending in a dangerous cross moments later.
Despite Ipswich's dominance, the hosts manage to find the back of the net in the 76th minute, as Shane Ferguson found Burke with a well-placed cross.
Scott Allan almost put Birmingham ahead just over a minute later, but his strike slipped past the left-hand post.
Lee Clark's side pressed hard for a winner late on, but some solid defending by Ipswich ensured that the two clubs shared the points.