The Italian side had led Group A in the early stages of the competition, but defeat in the capital of the republic of Dagestan saw them only with a mathematical chance of going through.
Anzhi went into the fixture on the back of some unconvincing domestic form which has yielded draws against Lokomotiv Moscow and Rostov, but Guus Hiddink's side answered their critics with a comfortable win.
Udinese keeper Zeljko Brkic easily dealt with a free-kick from Anzhi's Mbark Boussoufa after Rasim Tagirbekov had been fouled by Willians as the home side pressed for an opening goals in the opening stages.
Samba then hit a shot from distance, but his effort went wide of the target.
Boussoufa hit a shot wide of goal as the home side continued to dominate the early proceedings.
Brkic was on hand to parry a shot from Oleg Shatov as Anzhi looked for the opening goal.
Udinese's Diego Fabbrini shot from distance, but Anzhi keeper Vladimir Gabulov was equal to the effort as the visitors at last created an opening.
Eto'o nearly gave his side a 25th-minute lead only for his shot to be pushed round the post by Brkic.
Yury Zhirkov's shot was deflected wide by Larangera Danilo, while Boussoufa's corner was headed wide from close range by Silva Jucilei.
Gabulov kept his clean sheet intact when he caught the ball from an Antonio Di Natale header from a cross by Pablo Armero.
With 20 minutes to go Samba gave the home side the lead when he scored from close range following an inswinging corner by Boussoufa.
Five minutes later Eto'o doubled Anzhi's lead when he shot low into the bottom left corner of the goal.
In the final moments Anzhi's Shamil Lakhiyalov tried to grab a third but his shot from 25 yards failed to find the target.