The teams were tied at 1-1 after the opening day in Zagreb and both nations sprang a surprise by including their top singles players for the pairs match-up.
It meant Marin Cilic teamed up with Ivan Dodig for the hosts while Argentina had Juan Martin del Potro partnering Leonardo Mayer.
The clash was cagey but the Croats continually got the upper hand, winning the opening two sets thanks to tie-breaks.
After claiming a vital break in the third at 4-2, they held on for a 7-6 7-6 6-3 win in front of a boisterous home crowd, and they will now take a 2-1 lead into Sunday.
Victory for Cilic over Del Potro in the first singles rubber of the final day will see the hosts win the Davis Cup for the second time in their history, having previously claimed the annual event in 2005.
Dodig admitted afterwards he had been inspired by the crowd and says his team will do all they can to help Cilic bring the trophy home.
"This is a once in a lifetime experience and I have to thank the fans for creating a fantastic atmosphere," he said in a courtside interview. "We will be rooting for Marin to finish the job tomorrow."