The Falcons are having an excellent season under Dean Richards and Enisei felt the force of the Premiership side at Kingston Park, where Adam Radwan scored a second-half hat-trick in a ruthless rout.
Newcastle have now won seven matches in a row and their impressive run was never in danger of being ended by the Russian side, who showed why they prop up Pool 1 by conceding 10 tries in a brutal hammering.
Tom Penny touched down in the opening minute and further scores from Joel Hodgson, Glen Young and Simon Uzokwe gave Newcastle a 24-0 lead at half-time.
Hodgson completed a double and there was a treble for Radwan as the Falcons moved into the last eight at a canter.
The Blues faced a far greater challenge to secure top spot in Pool 2, coming from behind to eliminate Toulouse.
Gareth Anscombe proved to be the match-winner at Cardiff Arms Park, taking a pass from Willis Halaholo to dart over and adding a late penalty after a Cheslin Kolbe five-pointer and eight points from the boot of Thomas Ramos twice put the Top 14 club in front in a repeat of the 1996 European Cup Final.
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