The Cameroon international struck twice in the second half to inflict a second successive defeat on their Swiss opponents.
Anzhi had played out a 1-1 draw with Udinese in their Group A opener, while Young Boys were beaten by Liverpool in an eight-goal thriller.
The visitors created the first chance at Moscow's Lokomotiv Stadium, with Alain Nef's header swept off the line by defender Rasim Tagirbekov after just seven minutes.
Anzhi soon settled into their stride and Joao Carlos drilled over from distance before Brazilian compatriot Jucilei fired wide of the left post from outside of the area.
Young Boys threatened again on the half-hour through Jan Lecjaks, who pounced on a poor clearance from Christopher Samba only to see his left-footed effort well saved by goalkeeper Vladimir Gabulov.
And they came close again four minutes into the second half when Gonzalo Zarate unleashed a flier from the edge of the area which narrowly cleared the crossbar.
However, it was Zarate's error that led to the opening goal after 61 minutes. The striker was penalised for a handball in the area and Eto'o made no mistake with the resulting spot-kick, placing his shot into the bottom-left corner.
Eto'o almost had another next to his name eight minutes later but directed his header from Mbark Boussoufa's corner over the bar.
Young Boys' attempts to get back on level terms saw both Nef and Juhani Ojala head the ball wide within the space of two minutes.
And they were made to pay for the misses when Eto'o wrapped things up in the final minute by turning in a cross from Oleg Shatov.