Connor stepped up to convert the spot kick, won for a handball by Derry defender Paddy McLaughlin, and put United back in front at 2-1, having been brought back level by City midfielder Davy Byrne's second Cup goal since the appointment of manager Stephen Kenny.
The tie began to turn in the Blues' favour as Dublin referee Paul McKeon sent Derry midfielder Gareth McGlynn to the line, ten minutes before half-time.
Seven minutes in and Daryl Murphy capitalised on an error by Derry's beanpole defender Clive Delaney, to hammer home his eighteenth goal of the season past former West Ham 'keeper David Forde.
While Waterford's tails were up for a while, they were brought back down to earth six minutes later as Byrne volleyed home from the edge of the United area, after McGlynn's free kick had been headed clear.
Murphy and Waterford defender Ben Whelehan then threatened for the visitors before Connor was given his chance to convert a 35th minute penalty for 2-1.
McGlynn walked for a late challenge on Waterford midfielder Mark Clifford, and with that Derry's challenge seemed dead.
Murphy headed just wide for the visitors on 71 minutes and while a fit-again Sean Hargan drove over for Derry, PJ Banville struck the home side's crossbar in the final moments.
Derry City: Forde, McCallion, Hargan, McLaughlin (O’Grady, 88), Delaney, Doherty, Byrne (Hutton, 80), Deery (Beckett, 72), McGlynn, Martyn, Murphy.
Waterford United: Connor, Coleman, Frost, Whelehan, Breen, Mulcahy, Reynolds, Carey, Clifford, Bruton (Banville, 73), Murphy.
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