Four people made their way on to the field with Croatia trailing 2-1 against France in Moscow.
Lovren pushed a male protester, helping a steward to detain him, and later revealed his anger as his side went on to lose 4-2.
"I really was mad because we'd been playing at that moment in good shape," he said.
"We'd been playing good football and then some interruption came. I just lost my head and I grabbed the guy and I wished I could throw him away from the stadium."