Celtic managed to carve out the first chance of the half on 15 minutes as Given Mashikinya sent a powerful effort just wide of the upright.
The Urban Warriors could have broken the deadlock in the 21st minute but captain Mosa Lebusa directed his header wide of the upright.
Stanley Menzo's charges eventually took the lead in the 37th minute as Mzwakali turned home Roscoe Pietersen's cross.
That proved to be the final chance of the half, as Ajax went into the break with a slender lead.
After a quiet start to the second stanza, the home side eventually doubled their lead on the hour mark courtesy of a strike from Morris.
Celtic were gifted a goal in the 67th minute when Ajax defender Isaac Nhlapo deflected the ball into his own net.
Menzo responded by introducing former Royal Eagles striker Sedwyn George for Prince Nxumalo with 18 minutes remaining. George, however, could not get his name on the score sheet.
The visitors pushed for an equaliser in the closing stages, but Ajax managed to hold out to end their three-game winless run.