The team made the announcement on Thursday, hours before Game 5 of the best-of-seven Eastern Conference quarter-final.
Budaj replaced Price for the overtime period in a 3-2 loss Tuesday night in Ottawa. He faced two shots and allowed a goal from Kyle Turris.
Price appeared to injure his groin late in regulation time. He has struggled at times in the series and had a 3.26 goals-against average and .894 save percentage.
The Canadiens announced Wednesday that captain Brian Gionta would miss the rest of the playoffs with a torn biceps tendon. He is to undergo surgery on Friday.