Several stewards reported minor injuries after a clash with Birmingham fans, with reports of those in the crowd destroying seats and throwing projectiles.
The hosts secured a narrow 1-0 victory, thanks to a 65th minute goal by Kalvin Phillips.
But violence erupted after the final whistle at Elland Road, West Yorkshire Police said, with fans goading each other as police and stewards sought to calm them down.
“Terrible scenes inside Elland Road from the Birmingham City fans,” BBC sports editor Jonathan Buchan wrote on Twitter. “A few stewards receiving punches and kicks as they tried to apprehend a Birmingham fan who ran onto the pitch.”
Birmingham City’s caretaker manager Pep Clotet said after the Championship match: “Of course you don't want violence in the game anywhere. I just hope that no-one is hurt and calmed down as much as possible. You come to football to enjoy it and have a good day.”
Officers separated angry fans outside the stadium during an incident lasting around 45 minutes until the crowds dispersed, a police statement said.
A public order incident occurred also in the coach park outside the stadium, a police statement said.
“Police resources were deployed outside Elland Road and officers made 11 arrests in total for offences both inside and outside the ground," said Chief Inspector Jon Arrowsuch, the match commander.
"Clearly these were scenes no-one would wish to see and we will be examining CCTV footage and working with both clubs as we investigate what took place.
Enquiries are ongoing and police are appealing for information.