Can a creature's sacrifice benefit be taken when it is sacrificed to Eldrazi Monument?
No.

Magic: The Gathering Comprehensive Rules:


  117.10. Each payment of a cost applies to only one spell, ability, or effect. For example, a player can't sacrifice just one creature to activate the activated abilities of two permanents that each require sacrificing a creature as a cost. Also, the resolution of a spell or ability doesn't pay another spell or ability's cost, even if part of its effect is doing the same thing the other cost asks for.