Liverpool drew 3-3 with Benfica at Anfield to progress 6-4 on aggregate, while City drew 0-0 at Atletico Madrid to win 1-nil on aggregate.
The Reds will now play Villarreal in the last four, while City play Real Madrid.
Liverpool remain on course for an historic quadruple, having won the League Cup, they meet City in the FA Cup this weekend and are just a point behind City in the race for the Premiership.
They started the day with a 3-1 lead, but were made to work by the Portuguese side.