The Scottish side were in the market for a new keeper, while Sarkic wanted a move as he would get first team experience.
With Villa now in the Premier League, it seemed as though he would not get that chance at his parent club next term.
Now he has completed his loan move for the 2019-2020 season.
Manager Gary Holt said on the official Livingston website: "Between myself, Davie and GK Coach Tony Caig we feel we have brought a fantastic keeper to the club in Matija, he hasn't played a-lot of first team football but comes to us highly recommended and we have done our research on the lad which all came back very positive.
"We feel that this move is a win-win for all parties: Matija will get some much needed game time and we are getting a GK which realistically would have been a good bit out-with our wage structure.
"Matija will come in and push for the number 1 Jersey vacated by Liam and between Matija and Ross I feel we are more than covered and I feel confident going into the season that we have 2 keepers that would be fantastic number 1's.
"Competition for places is extremely healthy and is no different from any other position at the club, they need to work hard and make the jersey their own.
"Throughout preseason and into the Betfred cup section the fans will get a chance to see both lads in action."