Former Canaries striker Robert Earnshaw was denied by a post late on at the City Ground.
Forest took the lead against the run of play 10 minutes before half-time when Dexter Blackstock converted from the penalty spot after Russell Martin had tripped Paul Anderson.
But Norwich, who twice hit the woodwork in the opening six minutes, levelled in the 42nd minute when Andrew Crofts scooped the ball over Lee Camp - once on loan at Carrow Road - from 12 yards.
Both sides had chances to take the lead in the second half with Earnshaw being denied a winner in the 89th minute when his shot hit the post.
Norwich, bidding for three consecutive victories in the Championship for the first time since February 2008, made a bright start and were twice denied by the frame of the goal.
First Chris Martin, back in the starting line-up after sitting out the Carling Cup defeat at Blackburn in midweek, turned Forest defender Wes Morgan before firing a shot against the crossbar.
Martin was left frustrated again a minute later as he saw his free-kick from just outside the penalty area come back off the post.
Forest took 17 minutes to conjure up an effort at goal and when it came Chris Cohen - on his 250th career appearance - cleared the target from 25 yards.
The visitors continued to carve out opportunities and after Chris Martin cleared the bar from 25 yards, Korey Smith forced Camp into a fine save on the half-hour.
But Forest opened the scoring against the run of play in the 35th minute. Russell Martin brought down Anderson and Blackstock beat City goalkeeper John Ruddy, who had saved the previous spot-kick he faced in the last-gasp home win over Swansea last Saturday.
However, Norwich got the goal they deserved three minutes before the break when Crofts finished from 12 yards after Forest failed to clear their lines.
Cohen fired another effort over the bar five minutes into the second half before Norwich came close to taking the lead, Andrew Surman's 58th-minute corner being headed over by Elliott Ward.
Luke Chambers headed over the bar from Cohen's cross and Chris Martin did likewise when teed up by strike-partner Grant Holt moments later.
Earnshaw tested Ruddy in the 69th minute before Holt was foiled by the feet of Camp as play immediately switched to the other end.
Wes Hoolahan fired over the bar from 20 yards after Forest failed to clear Surman's corner.
But Forest came close to netting a late winner when Earnshaw saw his shot hit the post and Nathan Tyson headed over in injury time.