The visitors took the lead after seven minutes through Feng Gang but Drogba levelled just two minutes later before adding a second goal at the 84-minute mark.
Team-mates Feng Renliang, Giovanni Moreno and Joel Griffiths also found the target in a win which lifts Shanghai to eighth in the table.
League leaders Guangzhou Evergrand suffered a 2-1 reverse at the hands of third-placed Guizhou Renhe.
Zlatan Muslimovic opened the scoring after 35 minutes and while Gao Lin equalised just before half-time, a Chen Jie strike 10 minutes from time sealed victory for the home side.
Xu Liange's 74th minute effort was enough to give Beijing Guoan a 1-0 win over Dalian Shide while Changchun Yatai and Henan Jianye fought out a 2-2 draw.
Marquinhos and Zhang Wenzhou gave Changchun Yatai an early lead but Isaac Chansa and Xu Yang both struck to give their side a valuable point and lift them off the bottom of the table.
Two Bruno Meneghel goals and one from Zhu Shiyu gave Qingdao Jonoon a 3-1 victory over Liaoning Whowin.