The informal EU summit - without the UK - hopes to set a new course for the European project.
"We have to show with our actions that we can get better," Merkel said.
The EU had to improve "in the domain of security, internal and external security, the fight against terrorism, the cooperation in the field of defence", as well as defence and jobs, she added.
Further comment came from French President François Hollande who, speaking at a news conference, said, "We must be lucid about the situation that Europe is facing. This is not just another crisis. This can be the crisis of [the EU's] very existence. This is why we need to give Europeans a clear vision for what their future can and will be."
His comments were partly endorsed by European Council President Donald Tusk who said, "What we need today is an optimistic scenario for the future, no doubts. But it requires a realistic diagnosis of the causes of Brexit, and its political consequences for all Europe."
Arriving at the talks, Commission President Jean-Claude Juncker said, "Today is about overcoming differences. We need unity for Europe to work".
The meeting, the first EU summit excluding the UK in 40 years, will focus on the future of Europe and issues such as the current migrant crisis.
Ahead of the summit, BusinessEurope and its 40 national member federations sent a list of key messages on "how to build a better Europe" and what they expect in future negotiations with the UK.
In a statement, it said, "The EU must restore unity and urgently recreate the conditions to be a prosperity and security zone. The strategy to do so should include both rapid deliverables as well as longer term solutions."
Elsewhere, former MEP and European Green party joint leader, Monica Frassoni, said, "Brexit talks should not remain within the strict limits of an inter-governmental negotiation. We have to support the efforts of those in the UK, like the Greens, who strive to keep as much as possible of the 'acquis communautaire', especially in the areas of environment, freedom of circulation and establishment.
"It is necessary to prevent the UK from becoming a wild, liberalist kind of country, which suspends the Charter of Fundamental rights, becomes sealed for migrants, offers free space both for racist attacks and for tax avoiding multinationals.
"After the referendum on Brexit, we must avoid falling into a self-fulfilling prophecy of disintegration. The 'ever closer Union' remains a relevant target and an indispensable tool of changing the world towards more peace, justice and sustainability."
She added, "We need to prevent that the EU slowly ends up in an irrelevant and quarrelsome gathering of national governments."
Tusk hopes to focus on areas that the 27 leaders can agree on: border security, counter-terrorism and moves to "bring back control of globalisation".
But EU officials are playing down expectations of results from the meeting at Bratislava castle, in the capital of Slovakia.
The one-day Bratislava meeting is set to be the first in a number of confidence-building meetings where a "roadmap" should be set up to culminate in a summit in March in the Italian capital Rome, when the 60th anniversary of the EU's founding treaty of Rome will be celebrated.
A meeting of the PES leaders was held in Bratislava ahead of the summit.
PES President Sergei Stanishev said: "We all regret the Brexit vote, but I am glad that the future of the EU is now officially on the agenda, so we can shape it based on our social democratic values. Let’s not give in to the conservatives' agenda. We must offer the Europeans jobs, opportunities, stability and certainty."
The PES leaders agreed that economic growth and boosting employment and economic well-being are an absolute priority and that quick, tangible steps are necessary to recover the citizens' confidence. They discussed Swedish Prime Minister Stefan Löfven's initiative to host a conference on the project of a social pillar for the EU. A series of high level meetings of the PES family have been planned for the next couple of months.
Italian Prime Minister Mateo Renzi, Austrian Chancellor Christian Kern and the Prime Minister of Malta, Joseph Muscat, all encouraged the aim of having more meetings with a practical outcome.
Stanishev added: "This is a wakeup call, a call to end the 'austerity only' policy and our political family is determined to take the lead. Europe needs to go back to its social democratic roots and we are the only ones who can make that happen."