Golobart's attempted overhead kick became his second own goal of the season when it gave Caley Thistle keeper Ryan Esson no chance on 31 minutes.
But the 19-year-old atoned seven minutes from time when he fired home from close range. It was just reward for Inverness, who dominated for long periods.
The own goal came within a rash of chances for the hosts, who may find a share of the spoils was harsh considering the number of chances they created.
Both sides needed maximum points to keep alive their hopes of a place in the top six ahead of the split and Kilmarnock had the extra impetus of wanting to impress ahead of next week's league cup final against Celtic.
Wolves striker Sam Winnall made his first start for Caley Thistle and Scotland Under-21 international Graeme Shinnie was restored to the starting line-up after the 3-0 loss at Dundee United on Monday night, with Nick Ross dropped to the bench and Kenny Gillet missing out completely.
Dean Shiels was suspended for Kilmarnock following his red card in the 0-0 draw with St Johnstone last time out. Danny Buijs, Matthew Kennedy, David Silva and Lewis Toshney came in for the visitors, while Garry Hay and Dieter van Tornhout were not included and James Fowler was among the substitutes.
Caley Thistle started brightly with Billy McKay firing narrowly over.
Kilmarnock responded well with Mahamadou Sissoko denied the opener by Josh Meekings' goal-line clearance.
Visiting keeper Cameron Bell made a terrific double save to keep out McKay and then Winnall and Shane Sutherland was next to test him when he connected with a McKay cross.
Greg Tansey forced Bell into another top-drawer save after McKay's lay-off just inside the box. Gregory Tade then fired over from 10 yards after a clever one-two with McKay.
The lack of a cutting edge proved costly on 31 minutes when Golobart scored his bizarre own goal.
The Spanish centre-back, on loan from Wigan, attempted a scissor-kick clearance from the edge of the box but miscued the ball, which sailed over Esson.
Tansey ballooned a chance over as Inverness tried to get back in their stride.
Bell saved a low shot from Golobart but the hosts could not find an equaliser before the interval.
Tansey shot wide from the edge of the area and Golobart headed over as Inverness showed more attacking intent after the break.
The momentum stopped when Golobart clashed with Bell as he pounced on the ball and play was stopped for over two minutes for the keeper to receive treatment.
Unperturbed, the hosts continued their hunt for a leveller with Winnall curling a low effort inches wide.
Tade was next to squander a chance when he shot over for Inverness and moments later the Frenchman's header was tipped over by Bell.
Winnall was next to fire off target when bearing down on goal.
Buijs hammered an angled attempt from outside the area high and wide with the visitors' first real chance of the second half.
Sutherland's rising shot from the edge of the area was just over before Caley Thistle captain Richie Foran missed a gilt-edged chance in the closing stages when his side-footed effort from 10 yards was blocked by Bell.
The keeper then tipped over McKay's shot but moments later, after he saved Foran's header, McKay found Golobart who smashed home from one yard to make it 1-1 in the 83rd minute.
Michael Nelson was picked out unmarked by Lee Johnson's free-kick but Esson saved the header as Kilmarnock could not find a reply.
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