Marquis was only in the line-up because the Lions had a depleted strikeforce but popped up at the death to stab the ball over the line with the Norwich players furious with referee Mark Haywood for allowing the goal to stand as they appealed for offside.
Former Manchester United trainee David Fox had put Norwich on course for their first win in three games when he produced an excellent finish 15 minutes from time, his first goal for the club.
The recalled Chris Martin had the first chance for City after nine minutes, his 20-yard drive being parried by Lions goalkeeper David Forde.
Millwall hit back and twice came close within the space of five minutes.
First, centre-half Danny Shittu, having stayed forward after Norwich partially cleared a corner, saw his 20-yard effort superbly tipped round the post by goalkeeper John Ruddy, who then kept out a close-range header from Marquis.
The home side continued to press for the opener as half-time approached.
Scott Barron fired over from 20 yards in the 40th minute while James Henry, who had a loan spell at Norwich in 2008, then failed to find the target from 12 yards moments later.
There was still time for Ruddy to produce another excellent save to keep out a firm header by Danny Schofield.
Millwall began the second half as they finished the first, with Ruddy again called into action to deny Marquis for a second time.
Grabban then unleashed a shot from just outside the box as the Lions pushed for the opener.
Norwich finally threatened the home goal but two shots in quick succession from Martin and Korey Smith were scrambled away by the Lions defence.
City boss Paul Lambert made a double substitution in the 65th minute with Matt Gill and Jackson replacing McNamee and Wes Hoolahan in a bid to find a winner.
Smith was foiled by Forde from 25 yards before the visitors did take the lead with 15 minutes left.
Millwall failed to clear a long Lappin throw and Fox reacted first to smash the ball into the top corner.
The home side immediately shoved giant defender Shittu into attack and he came close to an equaliser but put his header over from six yards.
Ruddy thwarted Alan Dunne from 25 yards and Henry lashed wide from similar range as Millwall attempted to get back on level terms, while at the other end Holt hit his 18-yard effort past the post.
But just when it seemed Norwich were heading for victory, Marquis rescued the Lions.