The video, which was leaked to US celebrity website TMZ, showed Solange lashing out at Jay Z in a lift after the Met Gala on 5 May.
"Jay and Solange each assume their share of responsibility for what has occurred," said a joint statement.
"They both have apologized to each other and we have moved forward as a united family."
The hotel employee who leaked the footage has been identified and dismissed, it emerged on Thursday.
New York's Standard Hotel said it was "shocked and disappointed" by the leak and said it would it would hand over "all available information to criminal authorities".
The employee was dismissed for "breaching the security policies of the hotel and recording the confidential CCTV video," it added.
The three-minute video shows Solange - herself an R&B singer - enter the lift with Jay Z, Beyonce and their entourage.
She then confronts the rapper, whose wife stands by without interfering.
A bodyguard holds Solange back, although she again tries to kick Jay Z. There is no audio on the recording.
The full statement was released by the family to the Associated Press.
It reads:  "As a result of the public release of the elevator security footage from Monday, May 5th, there has been a great deal of speculation about what triggered the unfortunate incident.
"But the most important thing is that our family has worked through it. Jay and Solange each assume their share of responsibility for what has occurred.
"They both acknowledge their role in this private matter that has played out in the public. They both have apologized to each other and we have moved forward as a united family.
"The reports of Solange being intoxicated or displaying erratic behaviour throughout that evening are simply false.
"At the end of the day, families have problems and we're no different. We love each other and above all we are family. We've put this behind us and hope everyone else will do the same."
Jay Z, his wife Beyonce and her sister Solange say they are a "united family" despite a fight seen in a leaked video.