BELGRADE, Yugoslavia, Jan. 11— Government and Communist Party leaders in the republic of Montenegro resigned today in the face of widespread protests over the economy and abuse of power.

The resignations followed two days of protests in Titograd, involving as many as 120,000 workers and students.

The Montenegro resignations grew out of the economic hardships, ethnic strife and worker unrest in Yugoslavia that also caused the nation's Government, led by Prime Minister Branko Mikulic, to resign in December.

Witnesses said thousands of people in Titograd danced, cheered and sang when the latest resignations were announced. They added that protesters shouted ''Montenegro has risen!''