The home side made the better start with Montenegro captain Vucinic testing Shay Given with an angled shot which the Republic keeper pushed out for a corner.
Milan Jovanovic got free of Whelan from the corner but the defender's header was well wide.
The Republic then had a sniff of a chance in a counterattack in the fifth minute but Stephen Hunt failed to find the charging Kevin Doyle.
Montenegro continued to have the better of possession during the remainder of the opening quarter without putting the Republic defence under major pressure.
Robbie Keane appeals that Radoslav Batak has handled
At the other end of the pitch, McGeady won a free-kick in a threatening position in the 26th minute but the winger's own delivery was far too near goalkeeper Vukasin Poleksic.
Kevin Kilbane, again wearing a protective mask, put his own defence under real pressure two minutes before the break by being dispossessed but O'Shea averted the danger by producing a superb block to deny Steven Jovetic.
At the other end, the Republic appeared to be denied a stonewall penalty a minute before the break when Batak clearly handled after he had failed to deal with a Hunt cross.
After McGeady and Hunt had wasted a couple of free-kicks in threatening positions, the Republic went close to taking the lead on 55 minutes when Keane's hooked snapshot was beaten away by Poleksic.
Uncertainly between Dunne and Reid led to the concession of a free-kick in a dangerous position in the 60th minute but Mirko Vucinic blazed over.
A poor Batak clearance gave Kevin Doyle a sniff of a chance on 69 minutes but the Reading striker pulled his shot across goal and his club-mate Hunt was unable to get a good connection on the follow-up.
The Irish had a scare after Vucinic outfoxed Kilbane in the Irish penalty area in the 73rd minute but Given produced a smart save at his near post.
McGeady squandered the Republic's best chance of the match on 76 minutes when he blasted his angled effort into the sidenetting when a shot across the goalkeeper surely represented the better option.
With the game opening up, Jovetic's deflected effort looked set to beat Given on 78 minutes but the Irish goalkeeper reacted brilliantly to tip the ball away for a corner.
Montenegro produced some late pressure but the Republic held on for a deserved point.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.