The result lifted Ukraine into the runners-up spot on 12 points from eight games, one ahead of Finland who slipped to third after a 2-0 home defeat by group winners France, who had already secured an automatic berth with a match to spare.
Ukraine dominated the first half, with Roman Yaremchuk twice going close while Zinchenko and Andriy Yarmolenko also missed chances before visiting goalkeeper Georgiy Bushchan clawed out a Dennis Hadzikadunic header at the other end.
Zinchenko broke the deadlock in the 59th minute when he tapped the ball into an empty net from three metres as a fine run and cross from the left by Viktor Tsygankov left the Manchester City player with a simple finish.
Substitute Dovbyk sealed Ukraine's berth in the March playoffs when he raced clear from the halfway line and steered his shot from inside the area past home keeper Nikola Vasilj in the 79th minute.
The pacy Dovbyk could have had another in the dying minutes as he weaved his way past two markers and Vasilj before Adnan Kovacevic cleared his shot off the line.