Smith, who was with City for three months in 2007, headed his side in front in the 20th minute with his second goal in as many games.
The Canaries dominated possession but only tested goalkeeper Jamie Jones on a couple of occasions as they failed to score for only the second time in 15 attempts.
Orient were one minute away from being knocked out by Droylsden in the second round but can look forward to a possible plum tie while Norwich have now progressed to the fourth round just once in eight seasons.
Smith made no mistake in the 20th minute as he gave Orient the lead, heading home a Dean Cox cross following a short corner.
City almost levelled within minutes but Aaron Wilbraham's header from Russell Martin's cross went wide.
Source: PA
Source: PA