The BBC Radio 4 soap was downloaded more than 700,000 times, leaving Radio 1 DJ Moyles in second place with 424,000 downloads.
The corporation's online radio service - which allows listeners to tune in to live programmes and download archives on-demand for up to seven days after broadcast - received record use last month.
The combined downloads and live listening add up to 134 million hours of radio consumed over the internet since the beginning of the year. Listening to stations live online accounted for 16.4 million hours. Radio 4 alone experienced a 25% growth in on-demand listening hours.
But the corporation's experiments with downloads have not all been plain sailing. It ran into criticism from record companies in July when it made the entire Beethoven symphonies available online, and has pledged to scale down its ambitions for a Bach season on Radio 3 next month.