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The Blues had to wait until the 35th-minute to break the deadlock with a Sergio Aguero penalty, but thereafter the floodgates opened with further strikes from Aguero and Leroy Sane in a blistering seven-minute spell before the break.
Gabriel Jesus scored a seventh to complete the Blues' biggest Champions League victory ever on a night to remember at the Etihad.
City will learn the last eight opponents this Friday when the draw is made.
Schalke, needing a two-goal winning margin or at least to match the first leg score in normal time, began confidently and almost caught City out with a couple of early counter-attacks.
The visitors, backed by 4,000 travelling fans, had shipped four goals in each of their last two Bundesliga matches and chances for the Blues would surely come sooner or later and on 14 minutes, one did.
Kyle Walker’s pass set Raheem Sterling away on the right flank and Saturday’s hat-trick hero picked out Sergio Aguero who saw his low shot hit the outside of the post.
Little of any note happened then until the 33rd minute when the Blues were awarded a spot-kick.
Ilkay Gundogan spotted Bernardo’s dart into the box but the Portuguese was held back by Jeffrey Bruma before he could connect, and the referee awarded a penalty.
Aguero’s ‘Panenka’ penalty sent keeper Fahrmann the wrong way and the Blues were on their way.
Three minutes later the prolific Argentine had doubled the lead.
Gundogan again intelligently played the ball into space behind the Schalke defence for Sterling to run on to and his clever back-heel was controlled by Aguero and then toe-poked through the keeper’s legs to put City within sight of the quarter-finals.
It was Sergio’s 27th goal of yet another magnificent campaign and
And the Blues virtually assured progress with a third in seven minutes as David Silva played a clever ball through to Sane who raced on to the pass before putting another through Fahrmann’s legs.
Game, set and match.
Gundogan and Danilo had been City’s stand-out performers in the opening period and Gundogan again showed his impressive range of passing as he set Sane away five minutes after the re-start, but the German winger slid his shot wide.
Sane then had the ball in the back of the net moments later but the effort was ruled out by VAR – which had been used or consulted on each occasion.
And VAR was at it again within minutes as City sliced through the Bundesliga outfit’s defence at will, and as Sane’s cross was thumped home by Sterling, the flag was immediately raised by the assistant referee.
This time, the technology was in City’s favour and the decision was overruled for Sterling’s 19th of the campaign.
Schalke, by now thoroughly demoralised, were being picked apart at will and it was no surprise when Bernardo added a fifth on 71 minutes, firing in Sane's cross off the post
And still the torment wasn't over - Sane was again the provider (his third of the evening) as he played through Phil Foden and the youngster kept his cool as he waltzed around Fahrmann before rolling the ball into the empty net.
City completed the biggest Champions League win in our history as Gabriel Jesus fired a low shot past Fahrmann from 15 yards to complete the rout.
One goal and three assists for the former Schalke favourite. Danilo and ilkay Gundogan both had good cases for star man honours, too.
City go into the last eight of the Champions League for the third time.
The draw will be made this Friday and both legs will take place in April.
City, still in with the chance of further domestic silverware, travel to South Wales to face Swansea City in the quarter-final of the FA Cup.
Then it’s a welcome two-week international break before the Blues travel to Fulham in the Premier League at the end of the month.
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