United, who lost to Rangers on penalties in the Hampden final last season, survived a bright start from their First Division opponents before former Dundee midfielder Robertson drove the winning goal home from just inside the box with 15 minutes on the clock.
United may have made their victory more comfortable with more luck and composure in attack and almost paid for not being more ruthless.
Only a great save by United goalkeeper Lukasz Zaluska from Stephen Glass deep into stoppage-time prevented the game going another 30 minutes.
United survived some early pressure from the experience visitors but that appeared to be alleviated when Robertson drove them into the lead.
Willo Flood's cross from wide on the right arrived at the feet of the big midfielder and as the Dunfermline defence hesitated for the first time, he drilled a powerful volley from 16 yards past Pars goalkeeper Paul Gallacher.
The Fifers reacted positively and moments later striker Andy Kirk's drive from 14 yards was blocked by Zaluska who was impressively quick off his line.
In the 20th minute, as the game opened up further, United midfielder Craig Conway moved forward and unleashed a tremendous right-footed drive from 30 yards which Gallacher did well to tip over for a corner, which, again, came to nothing.
There was no discernible panic in the Dunfermline ranks but another moment of danger arrived in the 27th minute when Paul Dixon's inswinging free-kick from the right found Lee Wilkie at the back post but the United skipper's cut-back found no takers.
It was Dunfermline's turn to go even closer moments later when Kirk's shot from the edge of the box, which appeared to be partly blocked by Wilkie's challenge, went spinning past Zaluska before hitting the outside of the post.
The tempo of the game dropped slightly as half-time approached but United had clearly gained a grip of proceedings and a minute from the break Pars defender Austin McCann did well to clear Dixon's dangerous cross from the back post after the United midfielder sped into the Dunfermline box.
Despite their one-goal advantage, United went looking for a second to make certain.
Dunfermline went close just three minutes after the break when midfielder Glass curled a shot from 25 yards which just cleared the bar.
In the 50th minute Roy O'Donovan might have settled the tie when he was released through the centre of the Pars defence by Flood but the big striker's lob over Gallacher from just inside the box landed on top of the net.
On the hour mark, after the visitors defended two corners in quick succession, Robertson's drive from 25 yards was deflected by Dunfermline defender Greg Shields for yet another corner but still the second goal remained out of reach.
The Fifers sensed that they were still in the cup tie and drove forward for the equaliser but the home defence looked solid although Zaluska made a meal of Nicky Phinn's drive from distance, the Polish goalkeeper theatrically punching clear.
Midway through the second half, Dixon took a short free-kick from Robertson and blasted a drive from 30 yards against Gallacher's post and Dunfermline survived again.
The Fifers rallied momentarily and in the 75th minute Zaluska brought off a fine point-blank save from striker Graham Bayne who had latched on to Kevin Harper's pass at the near post.
In the 84th minute Flood nearly doubled the home side's lead, only to crash his left-footed drive from 10 yards off the Dunfermline crossbar.
In the fourth minute of stoppage-time Zaluska produced a great save from the volley by Glass, but United survived the late pressure to book their place in the semi-finals which will take place next February.