The hosts went ahead through Nick Ross' 21st-minute goal but missed a plethora of chances to extend that lead, and they were punished when Borja converted James Drayton's cross to earn a point.
Inverness were without injured captain Richie Foran but started well, with Billy McKay and Andrew Shinnie both going close early on.
Kilmarnock defender Manuel Pascalli then came to the rescue when he got a vital touch to prevent McKay tapping in Shinnie's pass.
However, Inverness' pressure paid off when Aaron Doran rounded Ross Barbour and crossed into the box, where Ross got in front of Jeroen Tesselaar and headed home.
Shinnie went close soon afterwards while Doran had a shot spilled by goalkeeper Kyle Letheren as Inverness pressed for a second.
Inverness remained on top after the interval, with Letheren blocking a Gary Warren shot, Ross Draper heading wide and Graeme Shinnie curling an effort inches past the post.
Terry Butcher's side had a strong penalty rejected on the hour mark when Momo Sissoko felled McKay, while Graeme Shinnie twice almost gave Caley the second goal their play merited.
Inverness' profligacy proved costly in the last minute when Borja pounced for his fourth goal of the season, moving Kilmarnock into top six in the SPL table.
The game ended on a sour note for Kilmarnock with manager Kenny Shiels sent off in injury-time for apparent dissent, having just returned to the dugout following a four-game ban.