Aziz's resignation came after an alliance of opposition parties had for several months accused him of being biased towards the outgoing government of the Bangladesh Nationalist Party.
Aziz had already been sidelined following three months of leave in November, in response to opposition protests so that an interim government could hold general elections without him overseeing them.
S.M. Asaduzzaman, an election commission spokesman, said: "He called the president on Sunday afternoon and tendered his resignation."
"Over the past few weeks, the political situation has changed rapidly. In order to prevent ... an undesirable situation, I have resigned from the post of CEC."
Resistance and reform
Aziz, a former supreme court official, was accused by opposition parties of rigging polls in favour of the Bangladesh Nationalist Party-led government, which ended its six-year term in October.
The opposition parties pledged to boycott the polls, sparking a political crisis that saw Iajuddin Ahmed, the president and the previous caretaker government chief, and his cabinet stand down on January 11.
The new government led by Fakhruddin Ahmed, the former central bank governor, pledged to undertake a string of reforms to win the confidence of opposition parties, led by the Awami League.
The reforms are expected to include a new voter list, identity cards, transparent ballot boxes and the depoliticisation of the civil service.