A crowd of 20,128 watched these two West Yorkshire rivals play against each other for the first time since the hazy days of the Premiership in 2001.
And the cheer at the final whistle from the home faithful also served to illustrate the importance of the game.
David Prutton shot against a post after four minutes but the breakthrough came when City midfielder Paul McLaren was adjudged to have handled and Robinson coolly converted the penalty after eight minutes.
City had a 21st minute goal controversially disallowed after Joe Colbeck's low shot cannoned off Frazer Richardson and goalkeeper David Lucas in a congested goalmouth.
Nobody appeared to know the reason why the 'goal' was chalked off and City boss Stuart McCall was spoken to by officials.
Becchio replaced the injured Enoch Showunmi in the 29th minute and 12 minutes later the striker headed home a corner from Robert Snodgrass after Robinson had struck the bar from 25 yards.
Conlon punished a mistake by Paul Huntington by drilling the ball in from an angle in the 71st minute to give City brief hope but Leeds held on, weathering the storm to progress to the next stage.