A positive start from the Candystripes saw Rory Patterson with the first shot on goal and it got even better when Michael Duffy stunned the home fans and levelled the tie with a simple finish after Patterson's shot was parried by the home keeper.
However, the reaction of the home team was swift as they made it 1-1 just three minutes later.
They hit the net again midway through the half and carried that advantage into the break.
Any hopes of a City revival were dashed as Shakhtyor punished some slack defending by the visitors in the second half to score three times and canter into the next round.
Derry City: Doherty, McBride, Barry, Byrne, Ventre, Molloy, McNamee, Dooley, P. McEleney,