Kenwyne Jones’ late header secured the point the Potters needed to book their place in the knockout stages of the competition.
It was a classic game of two halves at the Britannia Stadium, with the visitors dominating the first period and deservedly taking the lead in the 28th minute when Oleksandr Aliyev’s shot was turned into their own net by a combination of Matthew Upson and Asmir Begovic.
But Stoke rallied after the break and finally found the equaliser nine minutes from time when Jones nodded in Jermaine Pennant’s delicious cross.
The night began with a minute’s applause in memory of Gary Speed, and when the game did start it was Kiev who managed the first decent effort on goal, Aliyev testing Begovic from 25 yards.
The Ukrainians put Manchester City out of the competition last season and are leading their domestic league, but their record on English soil is significantly less impressive.
Ten previous visits have yielded just a single point, from a draw with Arsenal in 1998-99, while they needed an injury-time equaliser to deny Stoke victory in Ukraine on match-day one.
Kiev were looking dangerous, though, and Aliyev tried another long-ranger, this time missing the post by some distance, while Oleh Gusev’s cross along the six-yard line evaded everyone.
Stoke had struggled to get going and their task was made significantly harder in the 28th minute when Kiev deservedly took the lead, although in fortuitous circumstances.
Aliyev’s shot from the edge of the box looked to be going wide until the unfortunate Upson got a toe to it and guided it into his own net.
The hosts had failed to register a meaningful effort on goal, and the crowd were growing increasingly restless.
In the 38th minute Kiev nearly doubled their lead as former Chelsea striker Andriy Shevchenko was played in by Aliyev in acres of space on the left of the area but crashed his shot just the wrong side of the post.
Begovic rather summed up Stoke’s performance with a woeful piece of control to give away a corner just before half-time but fortunately for him it came to nothing.
Stoke needed to improve rapidly in the second half, and they managed their first decent effort on goal in the 52nd minute.
Kiev failed to properly clear a free-kick and Wilson Palacios broke the offside trap to get on the end of Delap’s dinked pass but he could not release his shot straight away and Oleksandr Shovkovskiy managed to block.
A couple of minutes later they came even closer as Shovkovskiy flapped at a long throw and the ball bounced nicely for Pennant, who hit his shot perfectly but was denied by a Yevhen Khacheridi block on the line.
Stoke boss Tony Pulis sent on Ricardo Fuller for Salif Diao with 20 minutes left and he was into the action immediately following a slack piece of defending by Danilo Silva but was crowded out before he could get his shot away.
Kiev were camped in their own half, and with nine minutes left the Potters got the goal they craved.
A long ball into the box was flicked on by Jones to Pennant, and when he swung in a cross the Trinidad and Tobago striker was in the perfect spot to nod in.
That looked like the hard work done but the hosts were grateful to Begovic for preserving a point, the Bosnian pulling off a terrific one-handed save to deny Andriy Yarmolenko with virtually the last kick of the game.