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.
Norwich were last night given a boost when manager Paul Lambert committed himself to the club despite Burnley’s interest.
The Clarets approached their npower Championship rivals on Thursday for permission to speak to Lambert about their managerial vacancy.
City refused and Lambert has insisted he is staying at Carrow Road, saying: “I am happy here and that is still the case.
“I want to put the speculation to bed – I’ve never had any intention of going elsewhere and nothing has changed.”
Lambert made six changes to the side which drew 1-1 at Middlesbrough last Monday with Declan Rudd, Zak Whitbread, Wes Hoolahan, Chris Martin, David Fox and Anthony McNamee starting.
Norwich came close to taking the lead in the seventh minute.
Fox picked out Martin whose cut back to Aaron Wilbraham was intercepted by defender Ben Chorley, who nearly turned the ball into his own net.
Orient, though, responded and Smith forced teenager Rudd into a fine save to keep out his 18-yard volley.
Smith made no mistake with his second effort 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 Wilbraham’s header from Russell Martin’s cross went wide.
Fox fired wide from 20 yards and Hoolahan was denied by Jones as the Canaries searched for an equaliser.
Orient nearly doubled their advantage five minutes before the break but Jonathan Tehoue could not direct Charlie Daniels’ corner on target before Matthew Spring’s long-range strike failed to test Rudd.
Norwich should have equalised moments later with Jones turning Wilbraham’s shot away for a corner, from which the unmarked Leon Barnett headed over when well-placed.
Barnett nodded another chance, from a Fox cross, over on the hour before Lambert made a double change in the 64th minute with Grant Holt and Simon Lappin replacing Wilbraham and Adam Drury.
Moments later Martin fired a shot wide from 25 yards and McNamee curled one just wide as City continued to press.
Substitute Paul-Jose M’Poku blazed a shot – Orient’s first in the second half – high and wide in the 75th minute.
Deep into injury time, Jones saved from McNamee and Holt headed the resulting corner wide.