Four days after opening their account with a 2-1 win over Georgia, Giovanni Trapattoni's side were exposed to intense pressure at both ends of a tight contest.
Given had to make excellent saves to deny Montenegro skipper Mirko Vucinic after three minutes and then again after 74 minutes.
But the Newcastle goalkeeper saved the best for last when he acrobatically clawed away Stevan Jovetic's deflected 78th-minute effort to preserve his clean sheet.
Ireland had their chances too with Robbie Keane forcing a good reaction save from Vukasin Poleksic, and they might have left Podgorica with all three points had Estonian referee Sten Kaldma spotted defender Radoslav Batak handling in the box on the stroke of half-time.
In the event, an away draw against a side determined to make a nonsense of their place in the last group of seeds in the qualifying competition could hardly be regarded as anything other than a positive outcome.
Trapattoni had admitted in the run-up to the game that he might say he was satisfied with four points from the two games, but be lying. As he headed for the dressing room, he may well have been having a re-think.
Keane led his players out into a bear-pit of a stadium as a hugely partisan home crowd prepared to roar on their country to what they hoped would be the first competitive victory of their short football history.
Having seen them surrender the opportunity to do just that against Bulgaria at the death on Saturday, Zoran Filipovic's men set about their task with some relish.
Vucinic led the way as a lone striker with excellent support from hugely-talented 18-year-old Fiorentina midfielder Jovetic and Elsad Zverotic, and Trapattoni's side had to weather something of a storm in the early stages.
Given was in action within three minutes of the kick-off, pulling off a good save low to his left after initially coming from his line and then stopping as Jovetic played Vucinic in.
Defender Milan Jonanovic got his head to the resulting corner, but his glanced effort flew just wide.
The visitors were having to work desperately hard to force their way into the game with Steven Reid, who was a doubt before kick-off with a knee problem, and the hugely impressive Glenn Whelan scrapping for every ball in an attempt to protect the men behind them.
But as time wore on, Ireland started to dilute Montenegro's threat, and that was reflected in a series of long-range efforts from Vucinic and Nikola Drincic which failed to arouse Given's interest to any meaningful degree.
The game sprang to life as the half-time whistle approached when Kevin Kilbane almost collected an unwelcome memory of his 50th consecutive competitive appearance.
The 31-year-old was pinned back deep inside his own half by Zverotic, and as he tried to extricate himself, he slipped and allowed the midfielder to surge into the box and pick out Jovetic.
The youngster set himself before pulling the trigger, but John O'Shea got in a magnificent block to protect Given.
But the drama came at the other end in injury-time when defender Radoslav Batak made a mess of clearing Stephen Hunt's cross and appeared to handle, although referee Kaldma was unimpressed.
Hunt argued long and hard with anyone who was prepared to listen as he left the pitch at the break, but to no avail.
Ireland started the second half at a much higher tempo than they had the first with Hunt particularly prominent, although his final ball at times left a little to be desired.
But it was from another free-kick that they almost took the lead with 56 minutes gone.
Steve Finnan floated the ball into the penalty area, where Kevin Doyle challenge helped to divert it into Keane's path.
The Liverpool striker span and sent a firmly-struck, left-foot half-volley towards goal, but keeper Vukasin Poleksic was well-placed to beat his effort away.
Whelan fired wide and Doyle volleyed straight at Poleksic, but with Ireland committing men to the search for a winner, Jovetic was finding space with ominous ease.
Given came to the rescue twice within four minutes when he blocked Vucinic's near-post effort and then somehow managed to to turn Jovetic's deflected shot away after initially finding himself wrong-footed.
Substitute Simon Vukcevic gave Given cause for concern with a late drive, but the ball sped past the post as Montenegro's hopes of victory dwindled.