The 23-year-old broke the deadlock from close range after connecting with a low cross from Eddie Nolan on 18 minutes.
Winnall had threatened to open the scoring 10 minutes earlier when he dispossessed Brian Saah, only to be denied by a brilliant recovery tackle by the defender at the last minute.
Chris Lewington needed to be at his best to stop his side from falling further behind just after the half-hour mark. The Dagenham goalkeeper produced a fine point-blank stop to thwart Niall Canavan.
At the other end, Sam Slocombe has yet to make a save. The Iron, who are unbeaten in their last 10 games, have now gone 405 minutes without conceding a goal.