Former US Open champion Cilic gave Croatia the upper hand by winning Friday's opening singles rubber against Federico Delbonis before Del Potro levelled for Argentina with a four-set win over Ivo Karlovic.
Both Cilic and Del Potro were drafted in for Saturday's doubles with the former teaming up with Dodig to claim the first two sets in a tie-break before wrapping up victory after Mayer dropped serve at 3-2 down in the third.
Olympic silver medallist Del Potro, who has bounced back after two injury-ravaged seasons, must beat sixth-ranked Cilic in Sunday's reverse singles to keep alive Argentine hopes of a first Davis Cup title.
Karlovic and Delbonis would then face off in the deciding rubber.
Croatia defeated Slovakia in 2005 for the country's only Davis Cup title, while Argentina -- beaten by Spain in both 2008 and 2011 -- have finished runners-up four times.