The former Leeds defender bagged his first goal since December 2000 and produced a man of the match display at the back to shut out the first division’s most prolific marksman.
After seven defeats in a row City were desperate not to match their worst ever run in the 1932-33 season but things didn't look promising early on.
After 17 minutes David Johnson headed straight at Bradford keeper Aidan Davison and City struggled to make an impact until midfielder Michael Standing saw his volley fly over six minutes later.
Forest midfielder Gareth Williams pulled a shot wide soon after and Davison rode his luck when he fumbled a cross and full back Mathieu Louis-Jean lifted his shot over an unguarded net.
The miss proved costly as Molenaar shook off his defensive duties to head City in front after 42 minutes.
Young fullback Lewis Emanuel drove over a perfect left wing corner and the 33-year-old Dutchman rose unchallenged to power a downward header past keeper Darren Ward.
Two minutes later City almost doubled their tally when Emanuel fired in another teasing cross, but this time Standing's close range header bounced to safety off a post.
Forest's busy play maker Jack Lester had a great chance to equalise midway through the second half when a hopeful cross cannoned off Molenaar's legs, but deflected just wide of Davison's far post.
With Johnson still a threat City were anxious to grab a vital second and their leading scorer Andy Gray should have eased the nerves 25 minutes from time.
After being put through by Standing he forced his way past Ward but then missed a left foot shot from 14 yards with only Des Walker to beat.
As the tension mounted Johnson put a free header just wide with 20 minutes to go and City hung on to claim their first win since they beat Norwich on October 26.