The Chivas USA man was the beneficiary of Segundo Castillo's exclusion, with the latter failing to recover from a knee injury suffered against Mexico last month.
Minda was initially included in Reinaldo Rueda's provisional 30-man squad for the tournament before being axed.
But the 30-year-old, who is back in the fold, said he was elated at his late call-up.
"I'm very happy. Very happy for this moment," Minda told Capital Radio.
"(I) dreamed of going to the World Cup. (I have) worked hard for this.
"(Now) I want to get to work with great dedication, with great joy."
Ecuador will play Switzerland, Honduras and France in Group E at the Brazil showpiece.
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