The former Ipswich trainee had been on the field for a matter of seconds before he headed home Marcus Stewart's cross with his first touch of the match.
Neither side could find the breakthrough in an even first half, but it was Norwich that came closest to breaking the deadlock.
Exeter lost possession in midfield and Cody McDonald found himself put through on goal, but after he rounded Exeter goalkeeper Paul Jones, Steve Tully was in the right place to make a superb clearance off the goalline.
The Canaries went close again shortly after half-time when Simon Whaley let fly from 30 yards, but his brilliant drive was tipped onto the crossbar by Jones. However, with their next attack, Norwich went in front after 52 minutes as Jens Berthel Askou climbed above Matt Taylor to head in at the back post.
Exeter responded eight minutes later as Stewart crossed from the right and Logan rose to plant a fine header inside the upright to earn a deserved share of the spoils for the Grecians.