The Danske Bank Premiership hosts begin fairly brightly, with Chris Curran, Joe Gormley and Martin Donnelly bringing saves out of Nenad Novakovic.
But Debrecen threatened more and more as time went on and they hit the woodwork through Ibrahim Sidibe in the 56th minute.
As their opponents continued to make attempts on goal, Cliftonville offered little in response until the final 10 minutes, in which skipper Ronan Scannell and Curran were unable to convert decent opportunities.
Substitute L'Imam Seydi then did likewise for the visitors in the dying seconds.
Earlier, Welsh Premier League outfit The New Saints suffered a 1-0 defeat in Slovakia to Slovan Bratislava.
Marko Milinkovic and Juraj Halenar kept Saints goalkeeper Paul Harrison busy in the first half.
And he was beaten seven minutes after the break when defender Erik Cikos, who was on loan at Scottish side Ross County last season, rifled in.
Mike Wilde subsequently passed up a good chance to equalise as the away side finished empty-handed.
The second legs of both ties are scheduled to be played next Tuesday, July 22.