Daniel Ayala found himself in the referee's book with only 38 seconds on the clock for a poor challenge, but in less than a minute Middlesbrough had snatched the lead.
Adomah's well-placed cross found Kike and the forward took advantage of a lapse in concentration among the Cardiff backline to beat David Marshall.
Boro had chances to put daylight between the teams inside the opening 10 minutes as first Adam Clayton then Adomah went close to a second for Aitor Karanka's men.
Cardiff tried to pull themselves back on terms, but Cala's wayward header from Peter Whittingham's delivery was as near as the Bluebirds came in the early exchanges.
However, Boro appeared to be enjoying plenty of promising attacks through the threatening work of Adomah as he crossed for Jelle Vossen later in the half, only for the Belgian's shot to be blocked by an increasingly besieged City defence.
Boro may consider themselves fortunate to still have 11 men on the pitch after Damia escaped with only a yellow card for a rash tackle on Mats Daehli.