But the Redskins (2-4) responded with a final 12-play, 80-yard drive, capped by Helu's 3-yard score.
The running back finished with 41 yards on 11 carries and Alfred Morris chipped in with 95 yards on 19 tries as Washington won for a second time in three games.
Robert Griffin III completed 18-of-29 passes for 298 yards, two touchdowns and one interception while adding 84 rushing yards on 11 carries in the win.
Jordan Reed led all receivers with 134 yards with one score on nine catches.
McCown, who replaced the injured Jay Cutler in the second quarter, went 14-of-20 through the air en route to 204 yards as the Bears (4-3) fell for the third time in four contests.
Matt Forte picked up 91 yards on 16 carries, also scoring three times for Chicago.