The 33-year-old, playing at his fourth Nations Cup, saw his penalty saved by Danilo and scored the opener in a seven-minute spell of the first half, then headed another midway through the second.
After Drogba took his overall Nations Cup tally to 10, Yaya Toure completed the scoring with an exquisite free-kick in the 81st minute with the stadium already half empty after disillusioned fans filed out.
The Ivorians coasted home against the co-hosts, who surpassed all expectations by reaching the last eight despite a modest 151th place in the world rankings.
"The game was as difficult as we expected, Equatorial Guinea have had a great tournament," said Ivory Coast coach Francois Zahoui, whose team have not conceded a goal. "I'm pleased we scored three goals and with the way we played."
Equatorial Guinea's Brazilian coach Gilson Paulo, who only took over the side one month ago following the resignation of Henri Michel, said he had done the best possible job in a short time.
"I knew it would be a very difficult game against experienced players with a lot of quality," he said. "I'm happy with the job we did in a short time."
Facing Ivory Coast's experienced team was clearly a bridge too far for a side consisting mainly of players from the Spanish lower league and without a single local-born player in the starting line-up.
Even though the co-hosts were facing one of the favourites in a 15,000 capacity stadium, there were still empty seats at kick-off.
After a poor start to the game, Ivory Coast had three successive shots blocked before the ball fell to Max Gradel who volleyed wildly over the crossbar from close range.
The Elephants were awarded a penalty in the 29th minute when Ben Konate tripped Didier Zokora just inside the area.
Drogba shot low to Danilo's right but the Brazilian-born goalkeeper managed to get his fist to the ball, sending the stadium into raptures.
The Chelsea forward made amends seven minutes later when he pounced following a comedy of errors in the Equatorial Guinea defence. The ball was headed back to Rui who failed to control it, allowing Drogba to steal in, cut inside a defender and, despite slipping, score inside the near post.
Drogba finished off the home team's chances when he headed in Toure's free-kick in the 70th minute, silencing the stadium.
Toure completed the scoring by curling a free-kick past Danilo from 30 metres with nine minutes to play.
