Following Porto's 3-1 win over RB Leipzig in the later game, Besiktas need one point from their last two matches, at home to Porto followed by a visit to Leipzig, to reach the knockout stages for the first time.
Second-placed Porto have six points ahead of Champions League debutants Leipzig on four while French champions Monaco, last season's semi-finalists, are propping up the group on two points and have only a slim chance of avoiding an early exit.
Roared on by their fervent fans in the Turkish capital, Besiktas pinned Monaco back from the start but the visitors missed the best two chances before they scored against the run of play on the stroke of halftime.
Rony Lopez silenced the Besiktas faithful when he beat Fabri with a superb curling shot from 20 metres after the home team's keeper had denied Monaco's Portuguese striker and Belgian midfielder Youri Tielemans.
Besiktas were rewarded for their pressure in the 54th minute as Monaco left back Jorge hauled down Ricardo Quaresma in the penalty area and Cenk Tosun powered his spot-kick past Danijel Subasic.
Monaco nearly regained the lead in the 75th minute when Oguzhan Ozyakup cleared an Andrea Raggi header off the line with the visitors looking dangerous whenever they came forward.
Both sides threw caution to the wind in the last 10 minutes but neither were able to find a winner, with Monaco striker Balde Keita squandering two good opportunities.
The result extended Besiktas's unbeaten home run in European competition to 12 games, although they fell short of racking up a fifth successive win.
(Writing by Zoran Milosavljevic in Belgrade,; Editing by Ed Osmond)
