Celtic made the brighter start in a jam-packed Astana Arena and their Dutch defender Virgil van Dijk had two headers cleared off the line in the opening minutes.
Roared on by their fans, Shakhter grabbed an 11th-minute lead when towering Bosnian defender Nikola Vasiljevic headed on a long throw for Finonchenko to poke into the net from three metres.
Celtic pressed forward as Kris Commons rattled the crossbar with a piledriver from 30 metres midway through the first half.
But the Scots, European champions in 1967, were again caught cold in the 77th minute, when a deflected Gediminas Vicius shot fell kindly for Khizhnichenko to head home from close range.
(Reporting by Zoran Milosavlevic in Belgrade; Editing by Rex Gowar)
