The hosts trailed 2-1 from the away fixture but two goals from Soufiane Rahimi, a Kaku penalty and Kodjo Laba's late double ensured a 6-3 aggregate victory after Yokohama goalkeeper William Popp saw red just before half-time.
Al Ain, the oil-rich United Arab Emirates' most successful club, won the inaugural Champions League final in 2003 and were runners-up in 2005 and 2016.
Harry Kewell's Yokohama were attempting to become the fourth Japanese winners after Gamba Osaka, Kashima Antlers and three-time winners Urawa Red Diamonds.
The win will be all the sweeter for Crespo after the 2005 European Champions League final, when the ex-Argentina international scored twice for AC Milan before Kewell's Liverpool came back to win on penalties.
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