Of the six people stabbed Thursday, two were injured seriously, Israeli police spokeswoman Luba Samri said.
Police arrested Yishai Shlissel, Samri said.
Israeli police say Shlissel, an Orthodox Jew, was released from prison three weeks ago after serving a 10-year prison sentence for stabbing and wounding three marchers at a Jerusalem gay pride parade in 2005.
Prime Minister Benjamin Netanyahu said that the government would "pursue justice for those responsible" for the latest attack. The government fell short of calling it an act of terror, but Netanyahu did call it "a despicable hate crime."
"In the state of Israel, the freedom of choice of an individual is one of the most basic values," the Prime Minister said. "We must ensure that in Israel every man and woman lives confidently in any way they (choose).
"This is how we operate and this is how we will continue to operate. I wish a speedy recovery to the injured."
Video aired on Israel's Channel 10 shows what appears to be part of the attack 10 years ago: a man making quick slashing motions at people on a Jerusalem street, including one holding a rainbow flag, while others try to subdue him.
Channel 10 also aired video in the moments immediately after Thursday's attack showing several wounded people, including a woman -- her shirt bloody -- who was put onto a stretcher and into an ambulance. Another woman was shown bleeding from her arm as someone applied pressure to the wound.
"We came together today for a festive event, but the joy was shattered when a terrible hate crime occurred here in Jerusalem, the capital of Israel," President Reuven Rivlin said of the incident.
"People celebrating their freedom and expressing their identity were viciously stabbed. We must not be deluded, a lack of tolerance will lead us to disaster. We cannot allow such crimes, and we must condemn those who commit and support them."