The 21-year-old was carried off in the 54th minute at Galabank and is likely to require an operation which will keep him on the sidelines for a lengthy period.
Cole was making his first start of the season against Annan after coming on as a substitute in the wins over Elgin City and Clyde.
Rangers boss Ally McCoist told the club's official website: "It's not great for Darren. He has a really bad ankle with some heavy swelling and unfortunately the immediate signs are that he will need an operation.
"I don't want to pre-empt the situation but if it is an operation then we will do well to see him again this season.
"It's a real shame because had been injured for a while and had just come back. I could see both physically and mentally that he has grown up.
"It has come at an horrendous time for him but hopefully he will handle it and be ready for pre-season if that is the case."