Henry was forced into a change as early as the 12th minute as Stefan Jovetic could not recover from an earlier knock and was replaced by Sofiane Diop.
The first half proved to be a frantic affair, as both sides struggled to carve out chances in the opening 25 minutes.
Despite the early set-back, Monaco broke the deadlock on 31 minutes. Aleksander Golovin broke down the right flank before picking out Moussa Sylla, who calmly slotted home from close range.
Five minutes before the break, the home side managed to restore parity through Wesley, who rose above his marker to nod home into the bottom corner.
The second half started off slowly, but the visitors nearly went ahead after 58 minutes when Wesley turned provider to pick out Rezei, who sent his effort just wide of the upright.
Henry opted for a change with 15 minutes as Diop, who replaced Jovetic in the first half, made way for Ivory Coast international Jean-Eudes Aholou.
Both sides pushed for a winning goal in the closing stages of the second half, but neither could carve out a clear-cut opening as the spoils were shared on the night.
Tottenham Hotspur and PSV Eindhoven played to a 2-2 draw in their Group B encounter.
Hirving Lozano put the hosts ahead, before Lucas Moura and Harry Kane put Spurs 2-1 up.
The visitors had Hugo Lloris sent off, while Luuk de Jong netted the equaliser for PSV with three minutes remaining.