Hosting City started the tie the stronger of the two teams and went in front with just five minutes on the clock. Build-up play between David Silva and Carlos Tevez 20 yards from goal released Yaya Toure, who went around Jamie Ashdown to send the ball into the net from close range.
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Having taken the lead, the home side went in search of a second and it arrived 10 minutes later. Aguero was brought down by Tom Lees inside the area and the Argentine striker stepped up to send the resultant penalty into the roof the of the goal.
Further headed chances fell to Toure and Aleksandar Kolarov before the break, but neither was able to breach Ashdown's net.
The Leeds keeper was beaten for a third time though seven minutes after the restart by the returning Carlos Tevez. Aguero and Silva played a one-two for the latter to chip an effort goalwards, where Tevez was waiting to roll the ball in.
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Jack Rodwell came off the bench on the hour mark following a lengthy spell on the sidelines and the midfielder almost added a fourth for City six minutes later. He was first to meet a parry by Ashdown, but his headed effort bounced to safety off the crossbar.
Roberto Mancini's team did manage to add another goal with 16 minutes remaining though as Aguero scored his second of the match when he rolled the ball into the net via the upright following a through-pass from Silva.
The result means that City are the first Premier League side to reach the quarter-finals of this season's competition.