Marcus Berg struck first blood for Al Ain early in the encounter when he flicked in a corner at the near post.
But Rafael Santos Borre struck twice in five minutes to put the Argentines 2-1 ahead.
Six minutes into the second-half it was all square. Caio levelled after a slick exchange of passes with Tsukasa Shiotani.
River Plate had the chance to retake the lead. They were awarded a penalty when Mohamed Ahmad felled Milton Casco.
Gonzalo Martinez's powerful drive smashed into the crossbar and away from the area.
In an enthralling shoot-out, nine players converted their spot kicks until Al Ain goalkeeper Khalid Eisa saved Enzo Perez's effort.
"I hope not to be the Croatia of this World Cup because I hope to do better," said Al Ain's Croatian coach Zoran Mamic. "Croatia lost in the World Cup final and I want to win it."
Al Ain are the first side from an Arab country to reach the final since the inception of the competition in 2000. They will face either the Club World Cup holders Real Madrid or the Asian champions Kashima Antlers in Saturday's final in Abu Dhabi.
"I've been in the UAE for a couple of years," added Mamic. "But I think yes, that we are perhaps looking at the greatest success in football in this country. I'm proud to be part of this success. I hope the best is yet to come."