Morocco went through with a 1-0 win over Tanzania on Sunday and Cameroon held on to beat South Africa 2-1. They meet in the next round on Friday.
Díaz, who has quickly emerged as Morocco's star player, broke the deadlock in the 64th minute. He beat Hussein Masalanga at his near post when the Taifa Stars' goalkeeper probably expected a cross.
It was Díaz's fourth goal in four games and it set off celebrations of relief around Rabat's near 70,000-capacity Prince Moulay Abdellah Stadium, where the home team will play all its games as long as it keeps winning through to the final on January 18.
The Atlas Lions had dominated possession but were largely frustrated by Tanzania's disciplined and committed defence.
Morocco captain Achraf Hakimi made his first start of the tournament after recovering from an ankle injury and played the whole game.
South Africa were left to rue a host of missed chances early on when Cameroon's defenders looked as though they were playing together for the first time.
An injury to Cameroon's Darlin Yongwa allowed both teams to re-adjust before play resumed. Yongwa was replaced by Aboubakar Nagida in the 21st.
Cameroon improved, supported by most fans in the cozy 18,000-capacity Stade Al Barid.
Junior Tchamadeu got the opener when he side-footed the ball inside the far post in the 34th after he was left free when South African defenders failed to clear Cameroon's first corner.
The Indomitable Lions emerged with more bite after the break, with Aboubakar Nagida crossing for Christian Kofane to head the second goal in the 47th.
Kofane missed a good chance to seal the win before Evidence Makgopa pulled one back for a frenetic finale.
Senegal and Mali booked their places in the quarter-finals on Saturday.
