The 34-year-old midfielder, celebrating an African record 170th international appearance, equalised at the right end and later grabbed a controversial third goal for Egypt as they kept up their bid for an unprecedented third successive title.
After the game ended 1-1 in 90 minutes, Mohamed Nagui and Hassan netted in extra-time to extend Egypt's record unbeaten run at the Nations Cup to 17 games.
The champions next meet arch-rivals Algeria in Thursday's semi-final in Benguela.
"It was my fault for the first goal but the team showed great character in fighting back," Hassan told reporters.
"It is also what we did in our first game against Nigeria."
Hassan's 94th-minute free-kick, which was partially blocked by Cameroon goalkeeper Idriss Carlos Kameni and bounced back off the woodwork, was adjudged to have crossed the line even though television replays indicated otherwise.
Hassan's landmark game started badly when he flicked an inswinging corner from Achille Emana into his own net after 25 minutes.
Cameroon's hopes took a knock when defender Aurelien Chedjou was sent off for a professional foul with nine minutes left in extra-time.
Cameroon captain Samuel Eto'o refused to blame defeat on Egypt's disputed third goal.
"We have to admit Egypt are a formidable team," he said.
