Ayr should have restored their lead moments later when Scott Walker headed wide after Chris Aitken's free kick found him at the back post.
Grindlay in the home goal had to look smart to tip Grant Murray's cross from the right over the bar after the wind forced it towards goal on the half-hour mark.
The wet and windy conditions were making it difficult for both teams and the game settled down after a frenzied start.
Decent play from Gibson saw him beat two Ayr men, but his cross was caught in the wind and landed harmlessly on the roof of the net.
Gormley missed a couple of good opportunities to hand Ayr the advantage after the restart, his first effort blocked by the goalkeeper and his second nodded wide of the upright.
The Scottish Premier League side forged ahead after 54 minutes after good work by Gibson on the left saw the ball centred to Bryson, who finished from just inside the area.
Pascali scored the opening goal against Killie's Ayrshire rivals
Ayr immediately tried to haul themselves back into the tie and Bryan Prunty was agonisingly close to meeting a Stevenson ball at the near post and poking home.
At the other end, Grindlay was called into action to pull off a superb double block, firstly denying Garry Hay's shot before keeping out the rebound from Wright.
Jim Jefferies' side squandered an opportunity to move further ahead with 15 minutes remaining when defender Mark Campbell's rash tackle on Hay led to a penalty for Kilmarnock.
Russell's effort from the spot lacked precision and was saved by Grindlay to keep Ayr in the hunt.
It was a crucial intervention as, with second remaining, substitute Williams volleyed home from the edge of the box to force a replay on Tuesday 20 January after Rascle punched clear.
The striker's joy was such that he stripped to the waist and kissed referee Steve Conroy, who swiftly replied with a yellow card.
Ayr manager Brian Reid: "It was a magnificent team performance and the draw was the least we deserved.
"Financially, it's probably even better for the club that we got a replay, and maybe even for Kilmarnock as well.
"Even with our goalkeeper saving a penalty, I felt then we were due to get something.
"I think if we'd lost the game it would have been a bit of an injustice."
Kilmarnock manager Jim Jefferies: "I think Ayr tried to use the conditions to their advantage by knocking the ball quickly and trying to put us under pressure.
"We got caught up in it, and I stressed to the players at half-time that this is conditions to be brave, to take the ball down and get it passed and played. Craig Bryson, Willie Gibson and David Fernandez came into the game in the second half.
"We got a stonewall penalty, should have been 3-1 up, and I think if we'd got to that stage we'd probably have gone on to win comfortably.
"Ayr got a massive lift. They pumped balls into the box and when they got corners these things were very difficult to defend. They're just hoping it falls for them, and fall for them it did."
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