The Toffees had to make do without England striker Andy Johnson, who failed to recover from injury, while Leon Osman and Joseph Yobo were handed recalls.
Following a 1-1 draw at Goodison Park in the first leg, Metalist started the better and put the Everton defence under pressure from the off. Fifteen minutes in Tim Howard made an excellent one-handed save from Edmar's low shot and Joleon Lescott and the keeper then combined to blot out the rebound.
James McFadden thought he should have had a penalty, before Metalist took the lead as smart play through the middle allowed Edmar to drive clear and place his low, left-foot shot into the corner of the net past Howard.
Half an hour in Lasha Jakobia's long-range drive forced another save from Howard as Everton continued to be pegged back, before Osman threatened at the other end with a header from Steven Pienaar's cross.
McFadden belted a left-foot effort from distance narrowly over, but Metalist remained well on top.
Three minutes into the second half Everton earned a lifeline as the keeper punched clear under pressure from Yakubu and Pienaar's ball back into the danger area was turned home from close range by Lescott.
However, within four minutes Metalist were back in front on the night and 3-2 on aggregate as Olexandr Rykun's drive from 20 yards hit the post and Hicham Mahdoufi was on hand to knock the rebound into the unguarded net.
Twenty minutes from time McFadden saw his shot blocked by keeper Olexandr Goryainov, before Pienaar's deflected drive flew narrowly over from the edge of the area.
McFadden then struck a decisive blow, making the aggregate score 3-3 with another outstanding strike, burying his left-foot drive into the corner of the net from 18 yards.
Edmar fired over as Metalist went in search of a winner, but Victor Anichebe had the final say as he latched onto a weak backpass and kept his composure to bury a low shot to seal a 3-2 victory on the night.