Substitute Jordan James struck deep into injury time to save the Blues as the battle of the new managers ended all square with Luke Williams' Swansea at St Andrew's.
Harry Darling put the visitors ahead, before Siriki Dembele quickly equalised, only for Jamal Lowe to restore the Swans' lead after the break before Wales international James drove home in the 95th minute.
Darling also hit the bar, but Blues wasted several chances as the hosts ushered in the Mowbray era after Wayne Rooney's departure.
"We could have scored five goals with the chances we had in the first half," said Mowbray.
"There were chances right in front of the goals where we had shots blocked and then (Siriki) Dembele ran through at the start of the second half I didn't see Swansea having those chances.
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