Ethopia were then awarded a penalty with 24 minutes on the clock with Chisamba Lungu felling Seid in the penalty area, but the striker fired his effort straight at Kennedy Mweene.
Drama evolved late in the first period, meanwhile, with Ethopia stopper Tassew receiving his marching orders after clattering Lungu some 40 yards from goal. The goalkeeper was stretched off in the clash, with the referee awarding the red card as he left the pitch.
It got worse for Ethopia in first-half stoppage time, however, when Collins Mbesuma raced through on goal before coolly finding the back of the net to send Zambia into the lead.
Both sides struggled to create opportunities in the early stages of the second period, but Ethiopia levelled after 65 minutes when Adane Girma drove into the penalty area before striking an effort into the back of the net.
Mbesuma thought he had been awarded a penalty when he was challenged in the penalty area by Lungu, but the referee booked the Zambian striker for simulation, although replays suggested that his shirt had been pulled by the defender.
Zerihun Tadele came close for Zambia late on when he turned in the area before striking an effort on goal, but Collins Mbesuma was more than equal to the danger.
Zambia continued to look the most likely in the latter stages of proceedings, but they could not make the breakthrough as the points were shared in the opening match of Group C.