The 20-year-old made a string of second-half saves to deny the Montenegrins victory in Podgorica and keep alive Ireland's hopes of making it out of Group 6 in their bid to reach the finals of the European Under-21s Championship.
Noel King's men capped an encouraging five days after beating the Faroe Islands in Sligo on Tuesday with a 0-0 draw, and remain level on points with Montenegro behind group leaders Germany, although having played a game more.
Ireland made a bright start to the game and within 16 minutes, home keeper Danijel Petkovic had been forced to makes saves from Sean Murray twice and Anthony Forde.
However, as the half wore on, Montenegro started to work their way into the game, although McDermott was largely untroubled and the first half ended goalless.
The hosts returned in determined fashion and went close four minutes after the restart when Stefan Mugosa had a shot blocked, although Petkovic was called upon once again two minutes later as Aiden O'Brien took aim at the other end.
McDermott had to deny Mugosa in a lively start to the second half, and he found himself the busier of the two keepers as he repelled another effort from Vladimir Jovovic with time running down.
Luka Djordjevic might have won it for the Montenegrins 13 minutes from time but for another save by McDermott, who then kept out attempts by Jovovic and Djordjevic with the home side throwing every they had at the Irish.
O'Brien had a chance to snatch three points deep into injury-time, but Petkovic ensured his side emerged with at least some reward for their efforts.