David Malpass said the bank expects global economic growth to shrink by 5% this year as nations deal with the pandemic.
This has already led to millions losing their jobs and businesses failing, with poorer countries feeling the brunt.
"Millions of livelihoods have been destroyed and healthcare systems are under strain worldwide," he said.
"Our estimate is that up to 60 million people will be pushed into extreme poverty - that erases all the progress made in poverty alleviation in the past three years," Mr Malpass warned on Tuesday.
The World Bank defines "extreme poverty" as living on less than $1.90 (£1.55) per person per day.