Following an early period of Kosovo pressure, the visitors started to grow into the game, albeit without committing numbers forward as they tried to limit any potential counterattack opportunities.
That meant chances were at a premium, and Chiedozie Ogbene curled well over the bar after being picked out by James Abankwah.
That proved to be an important moment in the half as it sparked the home side into action, and they laid siege to Caoimhin Kelleher’s goal.
In fact, the Brentford goalkeeper was helpless just before the half-hour mark as Muriqi smashed an effort towards the bottom corner, only to be denied by a sensational block from the retreating Jayson Molumby.
Another goalline clearance was needed five minutes later; this time it was Dara O'Shea coming to the rescue at the far post to turn Veldin Hodza’s goalbound effort to safety.
Heading into half-time, the Irish simply couldn’t get out of their own half, yet the game remained scoreless thanks to Kelleher when he clawed Muriqi’s looping header wide of the post.
The start of the second period saw a more end-to-end affair, with Troy Parrott heading straight at Arijanet Muric, whilst Hodza curled just wide at the other end minutes later.
With the game lacking any clear-cut chances, the hosts finally managed to craft a major opportunity to grab the opener on 66 minutes.
Baton Zabergja showed powerful running on the left to outmuscle Jake O'Brien before driving into the box, but his cutback was smashed high and wide by Edon Zhegrova.
With the game seemingly destined for a stalemate, the captain Muriqi stepped up to bravely nod home the winner from close range after Molumby had deflected a shot onto the underside of the bar.
The defeat extends a worrying away record for the Republic of Ireland, with only one win from their previous seven (D2, L4), and their next fixture against Israel won’t be played on home soil either.
On the other hand, Kosovo are mightily impressive at home with eight wins from their previous 11, but their credentials in the group will be tested with a trip to Austria next time out.
Flashscore Man of the Match: Vedat Muriqi (Kosovo)