County led through David Cancola's stunning first-half injury-time strike and had been on course for all the spoils until Ayo Obileye headed an 89th-minute equaliser.
County remain in the cinch Premiership relegation play-off spot but have only been beaten twice in their last eight games, and both of those setbacks came to the Old Firm - including last Wednesday's loss to Celtic when the Bhoys needed an injury-time winner.
County face St Johnstone in this Wednesday's crucial bottom-of-the-table clash and Mackay is delighted with the momentum his team are building.
Mackay said: "There are players beginning to understand how we're going to play.
"They are all new and have all been thrown together and they are starting to get there in terms of a pattern of play.
"At the start of the season 16 left and 12 came in and they don't know each other, they don't know me.
"I've got to install a way of playing, the fundamentals I want and they have to go and implement it.
"To be fair, over the piece this season we've acquitted ourselves not bad.
"We've lost two in eight and those are to the Old Firm and what we've got is a group who will run hard.
"And we've got the ability to score against any team in the league, as we've shown against the Old Firm.
"Any ego is left at the door, they come in and work hard for each other they are no cliques, I don't want that.
"I've got a group here now who are really tight with each other."
Livingston manager David Martindale enjoyed having a larger than usual crowd of 2,554 even though the club had hoped to play in front of nearly 5,000 after giving away 3,600 tickets to local children.
Martindale said: "I thought it was a good atmosphere.
"The kids aren't coming away with a defeat from their first game.
"To get over 2,000 after giving out 3,600 tickets, that was maybe signs of the Covid situation.
"It was a nice atmosphere and we were chasing the winner and got the equaliser.
"I personally enjoyed it."