Nick Ross took advantage of some hesitant defending to finish off Aaron Doran's cross midway through a one-sided first half but Inverness could not get the killer second and had a strong penalty appeal rejected when Momo Sissoko felled Billy McKay.
And Borja sickened the hosts when he shot low into the net following a low cross by James Dayton to earn his side the point that moved them into the top six of the Clydesdale Bank Premier League.
Shiels was back in the dugout following a four-match ban but he was sent off in injury-time for apparent dissent.
The Northern Irishman had used his rediscovered freedom to go into the away end before the game and shake the hands of the small group of visiting fans who had braved a snow-hit A9 to follow their team.
There was only heavy rain and a swirling wind when they arrived but their team were soon hit by a blizzard of chances as Inverness set about ending a four-match losing streak, which included a William Hill Scottish Cup defeat at Rugby Park.
McKay should have given the hosts a fourth-minute lead but headed over from four yards after Owain Tudur Jones had headed on Graeme Shinnie's corner, which came after Kyle Letheren spilled a cross.
Inverness, who were still without injured captain Richie Foran, soon threatened again when Andrew Shinnie powered a shot from 30 yards which Letheren dived to push away from the corner of his goal.
Manuel Pascali then saved Kilmarnock when he got just enough on Ross' drive across the goalmouth to prevent McKay having a tap-in after Andrew Shinnie's pass had cut open the visitors.
The pressure paid off in the 21st minute. Aaron Doran got round Ross Barbour on the left flank and whipped the ball into the six-yard box. Letheren stayed on his line and Ross got in front of Jeroen Tesselaar, who appeared to be waiting for his goalkeeper to collect, and forced the ball home from close range.
McKay was soon played in down the right but Letheren, who was deputising for injured Rangers target Cammy Bell, stood up to save at his near post.
Inverness remained on top Andrew Shinnie volleyed over following a cross from his brother and Letheren spilled Doran's powerful drive and was grateful to see the ball flash wide.
The hosts threatened from a series of corners after the restart. Letheren blocked Gary Warren's shot, Ross Draper headed wide and Graeme Shinnie curled inches wide of the top corner after his initial delivery had been cleared.
Kilmarnock had a major let-off 12 minutes into the half when Bobby Madden ignored a strong penalty claim. Sissoko lost possession to McKay and then made a clumsy challenge from behind as the striker ran into the box - but the referee somehow saw no foul.
The visitors brought on Borja, Danny Racchi and Rory McKenzie to increase their attacking threat. But, although they had more possession, they still struggled to carve out a chance.
Graeme Shinnie twice almost gave Caley Thistle the second goal their play merited after being set up by McKay, but he was just off target with his first effort and Letheren did well to save the second.
And the hosts missed the chance to move three points clear in second place when Borja netted the unlikely equaliser - and there was more drama as Shiels was sent off.
- Sports & Recreation
- Andrew Shinnie
- Aaron Doran