Assistant head coach Sergio Raimundo and goalkeeping coach Miguel Miranda have also left with immediate effect.
Former Watford, Cardiff City and Wigan Athletic manager Mackay is due to start work at Easter Road on 1 June.
The statement marking his recruitment said the 52-year-old will be in charge of the “entire football operation, encompassing all departments from recruitment through to the Academy. He will also lead on the appointment of the next head coach.”
Former captain David Gray steps in for a fourth spell as caretaker manager for the remaining two games of the season, with Motherwell visiting on Wednesday and a visit to bottom side Livingston on Sunday.
Asked why so many managers have failed at Hibs, Gray said: “If it was as simple as that, I’d love to do give you the answer. I don’t think it’s through a lack of effort.
“This is a great football club – you look at the facilities and everything it’s got, but that only counts for so much if you don’t find a winning formula on the pitch.
“We’ve underachieved this season and nobody is shying away from the fact that it’s been a collective – everyone needs to do better because we can’t just pin it on one thing.
“It’s a fantastic job for anyone that has the desire to manage at the top level in Scotland,” added Gray, who rebuffed questions about whether he would like the job permanently.
Leeds-born Montgomery, 42, spent most of his playing career with Sheffield United before moving to Australia with Central Coast Mariners and Wollongong Wolves.
He guided Mariners to the 2023 A-League championship after his move into management and succeeded Lee Johnson as Hibs boss in September.
He leaves Leith with a record of 12 wins, 12 draws and 13 defeats.
