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"What an incredible honor that you have given me," the Democratic presidential nominee told the crowd at the Wells Fargo Arena in Philadelphia. "And I can't believe we just put the biggest crack in that glass ceiling yet."
She added, "This is really your victory, this is really your night."
Clinton, who made the unannounced, brief remarks, also said: "If there are any little girls out there who stayed up late to watch, let me just say I may become the first woman president but one of you is next."
Earlier on Tuesday, delegates officially nominated Clinton as their party's nominee -- making her the first woman in U.S. history to become the nominee of a major political party.
Clinton's video remarks followed a lengthy speech by her husband, Bill Clinton.
Hillary Clinton will deliver her own speech at the Democratic Convention on Thursday.