Although a deserved victory, the elimination was harsh on the Third Division side who seemed to have secured a replay when goalkeeper Ashley Bayes dived full-length to deny Sergei Rebrov just a few seconds earlier.
From the ensuing corner, though, utility man Doherty - playing as a defender on this occasion - was found unmarked by Darren Anderton's inswinging cross and headed past Bayes.
The manner of defeat will have been especially disappointing for Orient who will have been thinking about a lucrative replay at White Hart Lane as the game drifted into injury time.
In truth though, Tottenham had all the chances and should have been at least a goal ahead at the interval as they dominated possession.
Orient's main danger came from the cultured left foot of Matthew Lockwood. This was first evident after eight minutes when a corner was cleared to him at the edge of the Spurs box and the young full back did well to angle a fierce volley arrowing for the roof of the Spurs goal, but Sullivan kept the effort out well.
After a scrappy opening, Spurs began to garner control of the game and had the ball in the back of the net after ten minutes, although Sol Campbell's effort was disallowed.
The England international hit home from close range after Bayes had parried his initial header, but he was adjudged offside from Ledley King's flick-on of Darren Anderton's cross.
After 20 minutes the Premiership side had another chance as Anderton curled in a left footed cross from the right flank that Ledley King met unmarked, but could only head straight at Bayes.
Five minutes later, it seemed that Tottenham - who are notoriously lucky when the year ends in 1 - may have been set for 'one of those days' as they hit the post twice within seconds.
Oyvind Leonhardsen clipped in a low cross from the right wing that King again met. On this occasion his effort had Bayes beaten but came back off the inside of the post. The onrushing Steve Clemence stuck a leg out at the rebound but saw his strike clip the crossbar, before going over for a goalkick.
With Spurs dominating possession, on the half-hour mark, Rebrov beat a defender in the box and tried a similar effort to the one he scored against Newcastle on Wednesday, but this time his shot goes over the bar.
Ten minutes before the break, Orient had Sullivan beaten for the only time in the game when Andy Harris met Lockwood's cross and headed against the frame of the Spurs' goal - although the effort would not have counted anyway as an offside was given against Okpara.
Five minutes after the break, Spurs came closest to scoring when Anderton's right wing cross was met by Rebrov's diving header at the back post, but Bayes parried away brilliantly.
In the last half-hour though, it looked like Orient had weathered the storm, with Spurs' only near miss coming when Tim Sherwood played Leonhardsen down the right hand side of the penalty area and the Norwegian hit a cross-shot that flew just past the Orient far post.
Indeed in the final stages, territorially Orient were having the better of the play and Harris' deep throws were a constant thorn in the Tottenham side.
To Spurs credit though, they held out well with Sullivan, in particular, dealt with the crosses flying across his box imperiously.
So Tottenham survived a potential Cup banana skin, against an in-form side and an inspired keeper, all of which may persuade Spurs fans that their great Cup record in years such as these will continue this season. George Graham, though, was more pleased to have finally secured an away victory.
"Its been a long time coming," Graham said. "The players knuckled down. If we'd scored in the first half I think we would have won more comfortably.
"We dug in and coped very well with their balls and Sullivan hardly had a shot to make all game."
Tommy Taylor was delighted with his side's efforts, but understandably devastated to have lost in these circumstances.
"It's a tough one, Andy Harris, who'd been superb all game, has let the runner go," Taylor said.
"I thought we did ever so well today. It might not have been a great cup tie, but we did well."
