Mayer, who unexpectedly replaced in-form Juan Martin del Potro for the deciding match, propelled the South American nation into their first final since 2011.
Andy Murray, whose defeat by Del Potro in a five-hour thriller in the opening singles on Friday left Britain facing an uphill task, had pulled the hosts level at 2-2 by beating Guido Pella 6-3 6-2 6-3 in the first of the reverse singles.
But Evans, ranked 53rd in the world, could not finish the job and Argentina secured a final showdown with Croatia, who had earlier beaten France 3-1.
(Reporting by Pritha Sarkar, editing by Ken Ferris)
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