Khan, 28, scored 117-111, 117-111 and 115-113 on the three judges' scorecards to claim a fifth consecutive win.
The Briton now wants to try to end triple-welterweight world champion Mayweather's 48-fight unbeaten run.
"Amir Khan wants to fight Floyd Mayweather," he said.
"I want that next. I'm number one World Boxing Council [challenger]. He's the champion. So let's make it happen."
Mayweather  beat Manny Pacquiao earlier this month in Las Vegas to add the WBO welterweight title to the WBC and WBA titles he already owned, in the most lucrative fight in boxing history.
Mayweather, 38, who is set to retire after a final fight in September,  turned down a meeting with Khan last year ahead of the Pacquiao bout, with the five-weight world champion preferring to  fight Argentina's Marcos Maidana instead.
Former two-time light-welterweight world champion Khan showed good stamina to overcome former light-welterweight world champion Algieri at the Barclays Centre, Brooklyn, scoring heavily with his quick combinations after a bright start from the New Yorker.
Algieri landed a strong combination of his own early in the eighth round, prompting a surge of cheers for the New Yorker from the partisan crowd, but Khan says it inspired him to dig deeper and see out the fight.
"He came to fight, wanted to win, had his home crowd, but I came here to win," he said.
"The crowd really fired me up the last three rounds when Chris really came at me hard."
Amir Khan wants to fight Floyd Mayweather next after claiming a unanimous points victory over American Chris Algieri in their welterweight bout in New York on Friday.