Having overcome more fancied doubles teams in wins over the United States and France en route to the final, Cilic and Dodig produced another outstanding performance to give Croatia a potentially pivotal advantage against Argentina.
"When trailing the Americans 2-0, we needed a miracle to stay in the tie and we've had a miracle that's propelled us all the way to the verge of lifting the trophy," Croatia coach Zeljko Krajan told a news conference.
"We know that Argentina won't go down without a fight and although they have no room for error, we expect an extremely tough final day."
The were no breaks of serve in the opening set but the Croatians dominated the tiebreak as Mayer faltered three times under pressure while the home team could not put a foot wrong.
Mayer dropped his serve in the third game of the second set and made amends to break Dodig's as the Argentines eventually forced another tiebreak, but the revival was short-lived.
The Croatians swiftly turned a 2-1 deficit into a 6-4 advantage with powerful serves and firm volleys before Cilic sealed the second set with an easy net point.
Cilic and Dodig again broke Mayer's serve in the sixth game of the third set and from there it was plain sailing for the Croatian duo, who closed out the match after Cilic unleashed a service winner.
Argentine fans were again led by World Cup winner Diego Maradona, who was wearing a black armband to mourn the death of his close friend, Cuban leader Fidel Castro.
Sunday's opening reverse singles features a battle of former U.S. Open winners as Cilic takes on Del Potro and Croatia's Ivo Karlovic faces Federico Delbonis in the potentially decisive fifth rubber.
Del Potro has won eight of his 10 meetings with Cilic, including the last five, but the Croatian was confident of coming out on top in what should be a mouth-watering contest.
"I expect a different match to those we have played recently as being ahead in the tie gives me a bit of a psychological edge," he said.
"Del Potro and I are in a similar situation as we've both spent two days on the court and fitness will be key because it could be a very long day."
Del Potro said: "Marin and I grew up together playing tennis. He is a really nice guy but tomorrow I face a big challenge in a must-win match against an in-form player.
"He has done really well (on the ATP tour) this year, the home crowd will be behind him all the way but as ever, I will do my best to win."
Croatia, the 2005 winners, are seeking a second title while quadruple runners-up Argentina are trying to win their first.
(Editing by Ed Osmond and Ian Chadband)
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