The 21-year-old missed Palace's trip to Turf Moor over the weekend, where Palace beat Burnley 2-0, and scans have revealed the injury will keep him out of the side's next four outings at least.
The issue is believed to have been picked up in training last Thursday and left manager Roy Hodgson without yet another option in the attacking areas.
Even though the upcoming international break will give Rak-Sakyi able time to make progress on overcoming the injury, a return is not likely until mid-December.
The club's Player of the Month for October will miss their Premier League fixtures with Everton and Luton Town this month, and most probably the early-December outings against West Ham United and Bournemouth.
Palace then face Liverpool and Manchester City in the run-up to Christmas, as well as rivals Brighton & Hove Albion, so having Rak-Sakyi back in time for that run of games will be a high priority.
Just as the club was able to call upon Matheus Franca after his summer move, and welcome back Eberechi Eze after a month out, the Eagles lost another player to injury.
Michael Olise also remains out for the Eagles, and despite taking part in some training exercises of late, a return is not scheduled until after the international break.
That meant that Jeffrey Schlupp had to fill in as a left-winger, more advanced than he has been used to of late, but he flourished, scoring the opening goal in the weekend win.
Hodgson was devastated to lose the youngster to injury on the eve of the game though, describing it as "a real blow."
The Palace boss added: "Just as you get Eberechi Eze and Michael Olise back almost into the fold, you lose another player who we think is very important for us in our attacking play."
Rak-Sakyi was Charlton Athletic's Player of the Year during a loan spell with the League One club last season after scoring 15 goals for the Addicks.
Eze played half-an-hour at Burnley though, and he is in contention to make his first start since the late-September win at Old Trafford versus Manchester United when Palace welcome Everton to Selhurst Park on Saturday.