The clearer rules for a union are also a partial victory for a group led by Germany and the Netherlands. Another group of countries led by France had argued that the rules should allow for more flexibility in deciding how losses would be apportioned and what creditors should be protected. Germany didn't want bailout funds being used to protect politically influential creditors.
The grand bargain that brought the ministers together preserves some flexibility for national authorities to protect creditors so long as shareholders and bondholders representing 8 percent of a bank's total liabilities are wiped out. It's a sort of minimum pain requirement. It also restricts the use of national funds for bailouts to a total of 5 percent the bank's liabilities.
This part of the union won't go into full effect until 2018, so it won't have any immediate effect. It's a solution for the next crisis—not this one.
"If the banks get into trouble we will now, throughout Europe, have one set of rules on who pays the bill," Dutch Finance Minister Jeroen Dijsselbloem told CNBC.
(Read more: Europe seeks to shield taxpayers from bank collapses)
As part of the banking union plan, depositors holding 100,000 euros or less would be fully protected from losses under the plan, much like depositors in the United States are protected by FDIC insurance.
The plan leaves in place the euro-zone's main bailout fund, the European Stability Mechanism. Last week the finance ministers agreed to an "operational framework" for recapitalizing banks with 60 billion euros from the fund in "extraordinary circumstances."
While national governments are required to contribute to the bailout, these contributions cannot go higher than 5 percent of the bank's liabilities. And the bailout funds are only supposed to be used to make depositors whole, not pay off bondholders. Use of the ESM funds would require authorization by the European Commission, the executive arm of the EU.
The hope is that the banking union will put an end to the so-called doom loop. That's the term used to describe a situation where sovereign debt troubles contribute to bank troubles and bank troubles contribute to sovereign debt troubles.
When a nation's finances looked shaky, people will become afraid that it won't be able to afford to support its banks if they get in trouble, causing people to lose confidence in the banks. And when banks are in trouble, that would put further strain on national finances, which would further strain banks, which would further strain national finance. And on and on it would go until national and financial collapse.
The combination of bail-in, limited national contribution and ESM should provide some relief from this vicious cycle since bailouts would no longer be limited to the wherewithal of national treasuries. That should mean a lot fewer days of watching markets around the world getting rocked by the latest worries about some bank you've never heard of in a country you mostly think of as a nice place to vacation.
(Read More: EU Strikes Deal on Who Will Pay for Bank Bailouts)
Last December, the finance ministers also agreed to a plan to grant the European Central Bank's supervisory authority over the largest 200 banks in the EU. Currently, banks are only subject to national supervision, which critics said created the danger that regulators might be too cozy with the banks they were supposed to regulate.
The banking union only partially solves this because smaller and mid-size banks will continue to be supervised by their national authorities. Pressure for to limit the ECB's authority came mainly from Germany, where regional banks are very politically influential and regulators plenty cozy with bankers.
Of course, this isn't quite over yet. The plan will require the approval of the European Parliament before it can become European law.
Bottom line: Europe is likely starting to climb out of the mess its been in for the last three years.