Britain had trailed 2-0 after the first day in Glasgow, but doubles victory on Saturday was followed by a singles win for Andy Murray on Sunday.
Dan Evans won the first set of the deciding match but Leonardo Mayer fought back to win 4-6 6-3 6-2 6-4.
Argentina will travel to Croatia for the final in November.
It is the fifth time Argentina have reached the final as they attempt to win the title for the first time.
Britain, who won the Davis Cup for the first time in 79 years 10 months ago, must wait for Wednesday's draw to see which nation they will face in the 2017 first round in February.
"It never feels good losing, of course it doesn't, but no regrets," GB captain Leon Smith told BBC Radio 5 live.
"We just asked the guys to go out and fight their hardest - the Argentines over the course of the weekend just played that little bit better and got the wins that were needed."
