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The 28-year-old from Ballymena needs one more victory in Rio to be sure of a medal.
It was a close contest but Donnelly, who won bronze at the Commonwealth Games in 2014, shaded it on a split decision.
He will fight world champion Mohammed Rabii of Morocco in the last eight.
That bout will take place at Riocentro on Saturday at 16:15 BST.
Thursday's contest was Donnelly's second at the Games as he outpointed Zohir Kedache of Algeria in his opening fight at Riocentro.
"There was not so much pressure on me as there was for the first fight and I think that showed in my performance," said Donnelly who finished with a badly-swollen left eye.
"I felt more comfortable as the fight went on.
"My next opponent is the world champion but I believe in myself and anything is possible at the Olympic Games.
"I have great belief in myself and am going to give it my all."
Ireland's Steven Donnelly has beaten Tuvshinbat Byamba of Mongolia on points to reach the quarter-finals of the welterweight division at the Olympics.