Leading tributes from the EU to the slain MP, Dimitris Avramopoulos said, "Extremism divides and nourishes hatred."
Cox, a 41-year-old mother of two young children, was the victim of a daylight attack outside her West Yorkshire constituency surgery by a man who, according to two witnesses, shouted "Britain first" during the assault.
She was a leading supporter of the UK's EU membership and took part in a high profile Remain event in London earlier this week.
Her death has stunned Westminster and led to the suspension of campaigning for next week's EU referendum until Saturday.
One EU insider said, "One question is just how this tragic event is going to play out in the remaining days of the campaign. Will the two sides lower the temperature? Or will it be business as usual?"
The far-right political party, Britain First, said it was "not involved".
The party's deputy leader, Jayda Fransen, said she was "extremely shocked" and added: "This is absolutely not the kind of behaviour we would condone."
Before her career as a British MP, Cox worked as head of Oxfam's Brussels office, spearheading campaigns for trade reform, and for former Labour MEP Glenys Kinnock.
It was the first killing of a serving MP since Irish Republicans murdered Ian Gow in 1990.
IMF chief Christine Lagarde refused to comment on Brexit questions "out of respect" for Cox.
Further reaction to her death came from S&D group Chair Gianni Pittella who said, "We are all greatly shocked and saddened by the news of the horrific murder. Jo Cox had spent time working with the Labour party in the European Parliament and was a powerful advocate of progressive causes. The thoughts of everyone in our group are with her family and friends at this terrible time."
Glenis Willmott, leader of the UK Labour delegation in the European Parliament, said, "We are deeply shocked and saddened at this tragic news. Jo dedicated her life to public service, as an MP, a charity worker and an activist."
Willmott went on, "She is fondly remembered by colleagues who worked with her in the Parliament and she will be deeply missed. It is simply incomprehensible that something so horrific could happen to Jo while she was working hard to serve her constituents. Our thoughts are with her family at this horrendous time."
Manfred Weber, Chair of Parliament's EPP group, said he was "shocked" by the murder, while Ukip cancelled a Brexit meeting on Friday "as a mark of respect."
Its MEPs Stuart Agnew and Patrick O'Flynn were due to attend the open public meeting.
Tory MEP Emma McClarkin said: "I am just stunned and devastated at the horrific attack on Jo Cox. My first thoughts and prayers are with her and her family."
"All parliamentarians stand shoulder to shoulder this afternoon, with heavy hearts and disbelief at this senseless attack."
Elsewhere, Lord Porter, Chair of the Local Government Association, said, "On behalf of everyone in local government I would like to extend my deepest sympathy to the family and friends of Jo Cox. Our thoughts are with them at this tragic time."
Nick Forbes, Senior Vice Chairman of the Local Government Association, said, "On behalf of Labour councillors I wish to express our deepest condolences to Jo's family. Jo was a committed campaigner and Labour politician who died carrying out her public service. We have all suffered a great loss today."