Romelu Lukaku's late penalty gave Everton a vital lead to take to Dynamo Kiev after the first leg of their Europa League last-16 tie.
A bad week for English clubs in Europe seemed to be getting worse when Oleg Gusev swept the Ukrainian side ahead.
The home side had started slowly but Lukaku's brilliant run teed Steven Naismith up to level before the break.
And the striker converted from the spot after Danilo Silva handled in the area to give the Blues a slender lead.
If they can hold it in Ukraine on 19 March, the Blues will set up a first European quarter-final in 30 years, an achievement set against the backdrop of a woeful league campaign.
Manager Roberto Martinez has conceded his side - six points above the relegation zone - face the harsh reality of a scrap for league survival but European commitments continue to offer respite.
Lukaku - much maligned for his league form in recent weeks - became the club's record European goalscorer with seven goals following his penalty.
Martinez will hope this hard-earned win can provide a platform for just a second league win of 2015 against Newcastle at Goodison on Sunday, while more optimistic fans may have one eye on a Europa League final appearance in Warsaw on 27 May.
Everton have made light work of sides well positioned in their own domestic leagues in the competition, including Wolfsburg and Young Boys, but it was not until Lukaku became heavily involved after half an hour that they started to trouble Ukraine's league leaders.
Gusev's near-post volley after he lost marker Ross Barkley saw Kiev take the lead as Everton began poorly, but when 40-year-old Oleksandr Shovkovskiy raced from his goal to allow Lukaku to shoot wide from 45 yards, the striker and his team-mates found their feet.
Lukaku, 21, quickly saw a powerful volley graze the crossbar and then showed strength beyond his years to fend off challenges and allow Naismith to curl home from 12 yards.
Everton did not look back and missed chances in the second half as they attacked the Gwladys Street end through Gareth Barry's miscue from 12 yards and Lukaku's own blunder from a Naismith flick in the six-yard box.
And they were rewarded late on as after substitute Leon Osman crossed against the hand of Silva, Lukaku's spot-kick squeezed past Shovkovskiy to seal a crucial win.
Oleg Gusev's near-post run allowed him to volley a third Europa League goal of the camapign
Steven Naismith's leveller arrived as Everton gained momentum late in the opening half
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