"Let's give each other a chance," he said at an event in Delaware addressing those who did not vote for him.
Mr Biden defeated incumbent President Donald Trump following a cliff-hanger vote count after Tuesday's election.
Mr Trump has yet to concede and has not spoken publicly since his defeat was announced while he was playing golf.
The result makes Mr Trump the first one-term president since the 1990s. His campaign has filed a barrage of lawsuits in various states but election officials say there is no evidence that the vote was rigged against him, as he has claimed.
Spontaneous celebrations erupted in major cities after media outlets announced Mr Biden's victory on Saturday. Disappointed Trump supporters demonstrated in some cities but there were no reports of incidents.