Twenty minutes from the end of a tense tie, the substitute sprung the offside trap to race clear — yet stroked the ball wide with goalkeeper Barry Murphy beaten.
It means the sides must replay on Monday, five days before they meet again in the EA Sports Cup final.
“We probably edged it on chances created,” said man of the match Daryl Horgan of Dundalk. “Nine times out of 10, Kurtis puts them away, so we’ll forgive him for that miss.”
Jason McGuinness wasted the first opportunity of note on five minutes when he peeled off Brian Gartland in the box to latch on to a cross from Kieran Marty Waters, but slipped at the crucial moment.
However, the Lilywhites began to show similar creativity to that displayed on their last visit to Tallaght by pressing their wide players, Daryl Horgan and Darren Meenan, right up on Rovers’ full-backs.
Richie Towell would have profited on 12 minutes only for a late intervention six yards out from Shane Robinson while, less than 60 seconds later, Horgan curled his 20-yard shot just a yard wide.
Rovers thought they’d grabbed a 26th-minute lead after Karl Sheppard rose high to head in Patrick Cregg’s deep free-kick, but the flag was raised for offside.
One of Towell’s trademark bursts from midfield almost had the visitors ahead 10 minutes later when Sean Gannon slipped the midfielder free, yet he was denied by Murphy’s save.
Rovers conjured little in the second half to show they were capable of ending their 28-year FAI Cup hoodoo. Indeed, as well as from Byrne’s sitter, Dundalk were close to sneaking it at the death when a floated free from Horgan forced Murphy into a save.
B Murphy; S Madden, J McGuinness, C Kenna, L Byrne; P Cregg, S Robinson; K Waters (Kilduff 89), R Finn, G McCabe; K Sheppard (D Kelly 67).
P Cherrie; S Gannon, B Gartland, A Boyle, D Massey; C Shields; D Meenan (Mountney 83), R Towell, D McDermott (K Byrne 54), D Horgan; D McMillan (P Hoban 77).
P Sutton (Dublin).