The Capital side came through a tough four sets to triumph 25-17, 18-25, 29-27, 25-20 and gain revenge for being knocked out by Mets last season.
The tie hinged on a pivotal third set but City held their nerve to edge it and get the momentum needed to go on and secure a quarter-final place.
City now meet Bellshill Cardinals, who beat Dundee University in straight sets, in the last eight.
Edinburgh University also made the quarter-finals with a straightforward 25-15, 25-12, 25-19 win at Stirling University but now face a testing tie against City of Glasgow Ragazzi who won at Bon Accord.
Edinburgh Jets tumbled out of the competition after losing in straight sets at home to South Ayrshire 15-25, 22-25, 19-25.