Drinan netted with a low drive into the bottom corner after the Finns skipper Omar Jama handled in a tussle for possession with midfieder Daire O’Riordan.
It was no more than the Leesiders deserved after they had the better of the first half action but couldn’t find a breakthrough goal.
Drinan had an early half chance and Kevin Taylor forced Robin Kallmann to a save in the opening ten minutes before the Finns Olander had an effort saved by man of the match David Coffey.
HJK were a big physical team but they were nearly undone after 25 minutes.
Taylor’s free kick reached Conor McCarthy and his header dropped just outside Kallman’s left hand post.
The Finns had two frees from Vesiaho and Malolo that went over and Coffey saved the second before Cork’s best chance of the half.
Kallman spilled the ball in the box in the 38th minute but Daire O’Riordan turned the ball wide of an open goal.
Alec Byrne clipped another shot over and then right on half-time O’Riordan was in again but his drive hit the ‘keeper and went up and over off the top of the crossbar.
The Finns missed a late chance to equalise and go through on the away goals rule when substitute Eetu Vertainen headed Vesiaho’s corner just wide and Coffey made a fine save from Olander in added time.
It was an historic win for Cork who now play AS Roma after the Italians beat APOEL from Cyprus 9-1 on aggregate winning the away leg 3-0 in Nicosia two weeks ago.
Cork City: Coffey; Taylor, McCarthy, O’Sullivan, Phllips; Fernandes, Coleman, Byrne, Ogbene; O’Riordan (Crowley 89, Leonard 91); Drinan (O’Regan 85).
HJK Helsinki: Kallman; Brockerman, Halme, Vesiaho, Olander; Ali (Ulmanen (46), Malolo, Jama (Vertainen 69), Stavitski (Kurki 85); Lappalainen, Ylatupa.
Referee: Petur Reinert (Faroe Islands).