The game, played on the plastic pitch at Moscow's Luzhniki Stadium, struggled to ever get going on a freezing night in the Russian capital.
Toon boss Alan Pardew made seven changes from the side that lost at Swansea last weekend, with Hatem Ben Arfa making his first start since December after recovering from a hamstring injury.
Anzhi boss Guus Hiddink named a strong line-up, with former Barcelona striker Samuel Eto'o, Brazil international Willian and ex-Chelsea duo Yuri Zhirkov and Lassana Diarra in the side.
Newcastle came with a plan to try and contain their hosts, who had won all of their home games in Europe so far this season.
The plan worked, with Anzhi restricted to just one shot in the first 45 minutes, former Cameroon international Eto'o firing in a dipping shot from 30 yards that Magpies keeper Rob Elliot did well to tip over.
Elliot had to be alert again 15 minutes into the second half when he got down low to his right to tip another Eto'o effort wide of the post.
Three minutes later Newcastle had a great chance to grab a priceless away goal when Ben Arfa found himself free in the area, but goalkeeper Vladimir Gabulov stood tall to parry away the Frenchman's chipped finish.
That proved to be Newcastle's only effort in a game of precious few chances, but Pardew's side will be confident of progressing through to the quarter-finals when the two sides meet at St James' Park next week.