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.
Match Reaction: Billy Davies praised his side's fighting spirit after the game while bemoaning the loss of two 'sloppy' goals.
"There's no doubt that we had one or two players that needed a game, including Chalobah and Reid.
"Chalobah has been injured, he needs minutes and games. He's like one or two of our young players who need education and they'll keep getting it.
"Chalobah is one who you can see has got a lot of quality and he just needs to get up to speed with the Championship and understand what it's all about."