Goals from Andy Reid and Nathaniel Chalobah twice brought the Reds back into the match in response to Millwall strikes from Morison and Woolford.
Forest's seventh draw of the season sees them remain in seventh place in the Championship ahead of another away game on Saturday at Sheffield Wednesday.
The managerless Owls produced one of the surprise results of Tuesday evening with a home win against league leaders Leicester City.
Source: Forest MAD
Source: FOOTYMAD
