Lionel Messi finished with a bloody nose and swollen cheek but Barcelona left unscathed, with one foot in the semi-finals and Manchester United hoping for another Champions League miracle.
The five-time European champions will face either Liverpool or Porto in the last four in their first appearance at the stage of the competition since 2015.
Manchester United needed two Paul Pogba penalties to stay in contention for a top-four Premier League finish by beating West Ham 2-1, while Tottenham moved up to third thanks to Lucas Moura’s hat-trick in a 4-0 win over Huddersfield on Saturday.
Ole Gunnar Solskjaer enjoyed a charmed life as interim Manchester United manager but Barcelona’s visit to Old Trafford on Wednesday will show him the scale of the task he faces as the new full-time boss.
Luis Suarez's 12th minute back-post header was deflected past David de Gea by United left-back Shaw.
Manchester City face Burnley on Sunday at the start of a four-match spell that could see Pep Guardiola write a remarkable new chapter in English football history.
Manchester United manager Ole Gunnar Solskjaer insisted he had no regrets over his decision to keep faith with David de Gea after another costly error from the Spaniard helped Chelsea claim a 1-1 draw at Old Trafford on Sunday.
The third act of Manchester City and Tottenham’s trilogy has some living up to do after a Champions League classic saw Spurs prevail on away goals to reach the semi-finals.
Manchester City moved back ahead of Liverpool to the top of the Premier League with a 1-0 win at Burnley, while Chelsea consolidated their place in the top four thanks to another error from Manchester United goalkeeper David de Gea in a 1-1 draw on Sunday.