The game appeared to be heading for a stalemate until the Canaries snatched a narrow advantage through a last-minute penalty from captain Cameron McGeehan after Josh Murphy was bundled over by Alex Davey.
Holders Chelsea started well and twice went close to breaking the deadlock in the first half with two efforts cleared off the line.
Nathan Ake's header was saved by City goalkeeper Will Britt and his second attempt was hacked away by Cameron Norman.
Jacob Murphy - Josh's brother - then fired just wide for Norwich before Islam Feruz almost struck for Chelsea on the stroke of half-time when his shot was headed off the line by Cameron King.
Jacob Murphy blazed over for the hosts after the restart but chances were at a premium in a tight second half with both sides appearing to have settled for a draw until the late penalty drama.
The second leg will be played at Stamford Bridge on Monday May 13, 7.45pm kick-off.