Morocco striker Mounir El Hamdaoui put the Dutch champions ahead three minutes into the second half before substitute Moussa Traore equalised in stoppage time just as Alkmaar's fans were preparing to celebrate a win over their Belgian neighbours.
Alkmaar, who had failed to score in their previous four matches, took the lead when El Hamdaoui turned and shot past Liege keeper Sinan Bolat.
Traore, who came on as a 77th-minute substitute, smashed the equaliser into the roof of the net from close range in the first minute of stoppage time.
El Hamdaoui could have settled the match minutes after his goal when a fine pass from Moussa Dembele set him up but Bolat blocked the effort.
The result left both teams with one point from their opening two matches in Group H. Arsenal top the group with six points, followed by Olympiakos Piraeus on three.