Trailing at halftime to an early goal scored by Ivan Schranz, Ukraine’s hopes were hanging by a thread but they produced a storming comeback in front of their joyous fans.
Slovakia, who shocked Belgium in their opening match, would have ensured a last 16 spot with a win but after a dream start they paid the price for sitting back.
Mykola Shaparenko equalised in the 54th minute with a close-range finish before they snatched a vital win as Yaremchuk controlled a lofted pass by Shaparenko to prod past Slovakia’s goalkeeper Martin Dubravka.
Victory lifted Ukraine level on three points with Slovakia and Romania, who face Belgium tomorrow.
Just as they did at Euro 2020, Slovakia had won their opening match in impressive fashion by stunning Belgium.
Three years ago, however, they still ended up crashing out at the group stage.
They began against Ukraine clearly determined to avoid a repeat and dominated the opening exchanges on a Dusseldorf pitch greased up pre-match rain.
Ukraine keeper Anatoliy Trubin, one of five changes to the Ukraine team beaten 3-0 by Romania, kept his side level with saves in quick succession to keep out a Lukas Haraslin shot and Schranz’s close-range effort.
Slovakia were rewarded or their positive start in the 17th minute though when Haraslin’s measured lobbed cross was headed in at the far post by Schranz for his second goal of the tournament, having also scored winner against Belgium.
With their tournament hopes looking as gloomy as the Dusseldorf sky, Ukraine finally sprang to life.
Mykhailo Mudryk blasted one shot high over the bar, had a better effort blocked by Slovakia defender Peter Pekarik before rightback Oleksandr Tymchyk thumped a skidding low drive against the post with keeper Martin Dubravka beaten.
Dubravka then had to be alert to beat out a freekick by Oleksandr Zinchenko, although at the other end Slovakia threatened to double their lead on the counter-attack with Haraslin’s shot well saved by Trubin.
Ukraine began the second half on the front foot and Artem Dovbyk, top scorer in La Liga last season, was just unable to direct a Mudryk cross at goal.
Kicking towards their yellow and blue-clad fans, Ukraine’s mounting pressure paid off a minute later when an unmarked Shaparenko was picked out by Zinchenko’s silky pass and he calmly sent a left-footed shot into the corner.
Slovakia looked content to hang on for a point but were fortunate when Mudryk struck the post with a shot from an acute angle with 15 minutes left.
Ukraine then dug deep and Yaremchuk was picked out by the excellent Shaparenko to send the war-torn nation into their final group game full of hope.