In front of the O's best home attendance of the season, 4,401, the visitors turned in a professional performance throughout.
Rovers took the lead in the 12th minute when Chris Brown pulled the ball back to the edge of the penalty area for Hume to shoot low into the bottom corner.
Doncaster doubled their lead nine minutes later after Orient had failed to clear a corner. Midfielder David Syers sent his shot into a crowded goalmouth where Cotterill applied a deft final touch from six yards.
Despite the greater corner count of six to one in the opening period, visiting goalkeeper Gary Woods was never tested although he was called into early action after the break.
Orient striker Kevin Lisbie twice forced saves from the goalkeeper, but the visitors; defence, superbly marshalled by giant centre-backs Rob Jones and Jamie McCombe, rarely offered their opponents hope.