The Scottish side eased to a 7-2 aggregate win against Jablonec in the third qualifying round, having seen its Champions League qualifying hopes dashed by Midtjylland, and was good value for its victory in Glasgow.
Furuhashi fired the host in front with 12 minutes played and teed up Forrest for Celtic's second with just over an hour on the clock, leaving the hosts in a commanding position ahead of the return game in Alkmaar next week.
AZ entered the competition at this stage by virtue of finishing third in the Eredivisie last season and the Dutch side, which has lost some key players since the end of the previous campaign, started the first leg strongly.
Zakaria Aboukhlal had a penalty appeal turned down in the opening few minutes and Joe Hart then produced a superb save with his toes to help Vangelis Pavlidis' shot onto the post.
But Furuhashi helped settle any Celtic nerves with his sixth goal in six games since joining, the Japan international athletically prodding the ball home after being picked out at the back post by David Turnbull.
Forest had a shot well kept out by Hobie Verhulst as Ange Postecoglou's side pushed for a second, though it was given a warning at the other end when Aboukhlal forced Hart into another save.
Shortly after Thijs Oosting had wasted a glorious chance for AZ by firing over from 10 yards, Celtic scored a potentially vital second goal through Forrest's shot on the turn, which took a wicked deflection off Timo Letschert on its way through.
Substitute Odsonne Edouard was twice denied by Verhulst in quick succession late on, but Celtic did at least keep out the opposition at the other end for an all-important clean sheet as it made it five wins in a row for Postecoglou.