Lukas Jutkiewicz fired the visitors ahead in the first half as he marked his return to the Riverside with a neat headed finish.
But Fletcher earned Middlesbrough a point as he converted Marvin Johnson's 81st-minute cross with a slick back-heel.
Boro enjoyed the better of a scrappy opening period, with Fletcher heading Djed Spence's cross wide of the target and failing to find the target with another driven effort.
However, when the home side were given a golden opportunity to take the lead midway through the opening period, they were unable to take it.
Gary Gardner conceded a penalty when he clipped Patrick Roberts' legs in the area, but Paddy McNair's spot-kick was a nice height for Lee Camp to deal with, and the Birmingham goalkeeper made a fine save tumbling away to his right.
Camp did not enjoy his time in the north east with Sunderland, with his spell at the Stadium of Light coinciding with the Black Cats' relegation to League One, but the 35-year-old will have relished his latest return to the region.
The same will have been true of Jutkiewicz, whose header fired Birmingham into the lead just three minutes after Camp was performing his goalkeeping heroics at the other end.
Birmingham came within inches of doubling their lead towards the end of the first half, with Kerim Mrabti drilling a low strike against the outside of the post after Aynsley Pears clawed David Davis' cross into his path to prevent Jutkiewicz tapping home a second goal.
Middlesbrough's problems continued with Roberts and Dael Fry both succumbing to first-half injuries – Jonathan Woodgate was already having to make do without Ryan Shotton, George Friend, Daniel Ayala and Britt Assombalonga – and Birmingham threatened again shortly before the hour mark as Kristian Pedersen looped a header onto the roof of the net after Bela crossed from the right.
Substitute Rudy Gestede came close to an equaliser with his first touch after leaving the bench, heading over from a corner, but Boro claimed a leveller with nine minutes left.
Marvin Johnson delivered a low cross from the left, which Fletcher converted via a neat back-heel from the corner of the six-yard box.
That looked like being that, but there was drama in the dying seconds as Jutkiewicz put the ball into the net moments after referee Andy Woolmer brought play to a halt because Marcus Tavenier was down a head injury.
The incident sparked a melee on the touchline that resulted in coaches Leo Percovich and Sean Rush both receiving red cards.