Swansea, Tottenham, and Wigan, with a first-ever European victory, all registered wins on Thursday to stay on course for the Europa knockout rounds.
Swansea made it two wins in two Group A games with its triumph over St. Gallen in south Wales.
Wayne Routledge netted his third goal of the season in the 52nd minute to settle the tie at the Liberty Stadium.
But St Gallen's Goran Karanovic had a first-half penalty saved by Gerhard Tremmel after Dwight Tiendalli had been penalised for handball.
"No game is going to be easy and they battled hard, and in the end it was a gritty win," said Routledge.
"The performance shows that we have massive character but in the dressing room we know that we can knuckle down and play if we have to."
Jermain Defoe celebrated his call-up to the England squad earlier in the day for this month's final batch of World Cup qualifiers with the first goal.
Set up by Lewis Holtby he struck from inside the box after the half hour mark, rifling the ball into the roof of the net.
That left the striker one goal shy of Martin Chivers' European club record tally of 24.
Nacer Chadli doubled up five minutes later for his second goal since arriving from FC Twente.
Again Holtby was involved, playing the ball out to Kyle Walker, with the right-back slipping the ball across to Chadli to sweep home.
The game, played on the Ramenskoye pitch in the Moscow suburbs 1,770km away from Anzhi's home stadium due to the conflict in Dagestan, turned more in Anzhi's favour after the break, but they were unable to find a way back.
Tottenham coach Andre Villas-Boas, who had fielded a nearly all-fresh line-up from the side that drew 1-1 with Chelsea last weekend and with West Brom looming on Sunday, looked more than satisfied with his 'B' side's showing as did the 150 or so fans who had made the trip from north London.
The Portuguese boss said: "In the second half we lost control of possession a little. However, defensively we were sound and managed to take a deserved win.
"A win is what we wanted and gives us the opportunity in the next games against Sheriff to ensure qualification."
Wigan marked their first ever home match in Europe with a 3-1 win over Maribor of Slovenia.
Nick Powell gave the FA Cup winners the lead with Ben Watson adding a second for a 1-0 half-time lead.