Falcons ran out 55-7 winners in the corresponding game at Kingston Park.
But the Russian side led 14-7 at the break as tries from Denis Simplikevich and Davit Kacharava sandwiched Calum Green's score.
Kacharava's second try and a penalty from Yury Kushnarev wrapped up a deserved victory for Enisei.
It was Enisei's second victory of the pool stages, while Newcastle have now lost three of their five games.
Falcons must now beat Brive at home next Saturday and hope other results go in their favour to stand any chance of reaching the last eight.
Enisei: Gaisin; Baranovs (c), Kacharava, Babaev, Simplikevich; Kushnarev, Shcherban; Volkov, Gasanov, Zykov, Saulite, Krasylnyk, Orlov, Temnov, Rudoi.
Replacements: Selskii, Polivalov, Pronenko, Sidorov, Kuzmenko, Serkov, Simonov, Butenko.
Newcastle: Penny; Kibirige, Powell, Socino, Agulla; Catterick, Booth; Rogers, McGuigan, Ryan, Green, Witty, Mayhew (c), Temm, Chick.
Replacements: Hawkins, Harris, Foster, Botha, Thompson, Dawson, Redman, Marshall.
Newcastle Falcons' hopes of reaching the quarter-finals of the European Challenge Cup suffered a blow as they lost 24-7 to Enisei-STM in Sochi.