Jack Lester netted his fifth league goal of the season to cancel out Williams' first-half opener but the striker struck again to give the visitors a deserved win.
Boa Morte, who joined the club on Wednesday, set up a good chance in the fifth minute when his corner gave Neal Trotman a clear opening but the defender headed high over the bar from eight yards.
Dagenham survived the early pressure and deservedly went in front three minutes before half-time when Williams was allowed to run at the defence and drive a low shot from 20 yards into the bottom right corner.
Chesterfield almost equalised from another Boa Morte corner in the 51st minute when Chris Atkinson hooked a shot against the bar but Lester made no mistake four minutes later when he turned in a Craig Westcarr shot at the back post. But Dagenham regained the lead two minutes later when Williams curled in a superb right-foot shot from the edge of the box after Chesterfield failed to clear.
Boa Morte was denied a debut goal when Chris Lewington rushed out to block his 67th-minute shot and the goalkeeper made another good save to turn behind a Terrell Forbes header to secure three points for the Daggers.