The game was goalless following extra-time after the Republic missed a series of glorious opportunities.
Conor Hourihane wasted the best Irish chance late in normal time with sub Alan Browne hitting the woodwork.
Browne then saw his penalty saved by Slovak keeper Marek Rodak before Matt Doherty’s miss sealed the home win.
It was a devastating conclusion to the Republic’s Euro 2020 qualifying campaign which saw Stephen Kenny taking over the managerial reins from Mick McCarthy heading into the play-off.
Kenny’s side undoubtedly deserved to progress to an all-Irish play-off final with Northern Ireland in Belfast on 12 November but were undone by a combination of poor finishing and bad luck.
If the Republic had progressed from the play-offs, they would have been guaranteed two home matches in Dublin in the new format multi-venue Euro Finals but now will have to watch from the sidelines as the competition comes to the Aviva Stadium.