Boro had been staring down the barrel of defeat after Paul Caddis put Blues ahead from the spot in first-half stoppage time, the visitors finding themselves a man down with goalkeeper Dimitri Konstantopoulos sent off for conceding the penalty.
But a storming second-half fightback earned its reward when on-loan Chelsea marksman Bamford struck for the 12th time this season in the 74th minute.
The draw may have ended Middlesbrough's run of six successive Championship wins, but a point was the least they deserved for their second-half display.
Bamford's equaliser denied Birmingham what would have been a morale-boosting win little more than 24 hours after their parent company, Birmingham International Holdings Limited, went into receivership.
Their winless league run has now stretched to six games - collecting just five points from 18 - as they wait to discover if there will be any fall-out from the Football League following the events in Hong Kong surrounding BIHL.
Middlesbrough's neat passing style saw them dominate the early exchanges, and it needed a fingertip save by Blues goalkeeper Darren Randolph to nudge away a low angled drive from Emilio Nsue after three minutes following a slick pass from Adam Forshaw.
But that was something of a false dawn for Middlesbrough as it was Blues who went on to carry the greater threat in the opening period.
Andrew Shinnie came close to giving them a ninth-minute lead when he let fly from 30 yards after good work from Clayton Donaldson, only to see his shot beaten away by Konstantopoulos.
Middlesbrough were struggling to deal with Donaldson and it needed a last-ditch block from Ben Gibson to prevent him testing Konstantopoulos from close range after 19 minutes.
Demarai Gray was then denied a 31st-minute opener by the intervention of the well-positioned captain Grant Leadbitter.
It was no surprise that Donaldson had a pivotal role in the breakthrough when it did come on the stroke of half-time
He was released by Shinnie and, after powering into the penalty area, was upended by a crude challenge from Konstantopoulos.
Referee David Coote had no hesitation in dismissing Konstantopoulos and, after Boro took off striker Jelle Vossen to send on Tomas Mejias, the first thing the substitute goalkeeper had to do was pick the ball out of the net as Caddis' penalty fizzed past him.
Blues looked certain to go on and race to a comfortable win but, when Jonathan Grounds and David Cotterill squandered excellent chances, Middlesbrough sensed the chance to force their way back into the game despite their lack of manpower.
Leadbitter had impressed all game for the visitors and it was fitting that it was his pass which teed up Bamford to beat Randolph from six yards.
Donaldson then had a chance to grab a dramatic stoppage-time winner from a Caddis cross, but headed over the bar from six yards to ensure that the spoils were shared