By the close, the FTSE 100 had gained 0.97%, or 64.57 points, to end at 6,737.95.
France's CAC-40 closed up 1.94% at 4,998.10 while Germany's Dax ended up 1.49% at 11,484.38.
The rises came after European Council President Donald Tusk said a bailout for Greece was "all ready to go".
US stock markets also traded higher early in the session.
Bank and financial shares surged, with Germany's Deutsche Bank shares closing up 3.3%, while the biggest winner on Frankfurt's main index was the stock exchange Deutsche Boerse up 4.38%.
However, after an early spike upwards, the euro fell 1.19% against the dollar to buy $1.1024, and tumbled 1.18% against the pound to buy 71p.
Nick Stamenkovic at RIA Capital Markets said: "The relief rally in the euro was short-lived as investors await details of the Greek agreement."
An initial drop in the yield, or cost of government borrowing, in Italy and Spain - the countries seen most at risk from the knock-on effects of a Greece exit - also tapered off.
Eurozone leaders have said that so far, a blueprint for a Greek debt deal has been agreed, with European countries' parliamentary approval processes beginning from Wednesday.
"For the markets, it's clearly positive that there is an agreement among the European member states and that there is an atmosphere of co-operation," said Philippe Gijsels, head of research at BNP Paribas Fortis Global Markets in Brussels.
"Still, there is a bit of execution risk which may haunt us in terms of volatility."

The gang carried a handgun, a crowbar and a machete as they burst into the Kirkcaldy home of Dwayne Kinner, who was with his cousin Nicholas Roberts.
James Mackie, 30, Derek Finlay, 31, Gordon Ellis, 24, and Scott Smith, 23, raided the flat on 8 January 2015.
A stolen phone with a tracking app took the police to the gang.
They also made off with a haul of Â£2,500.
The High Court in Edinburgh heard that the victims believed they were going to be shot after the gun was pointed at their heads and another robber threatened he was going to cut their fingers off.
The gun, which was discovered to be an air pistol, was later found behind a kickboard in a kitchen at a house in Glenrothes, which Finlay had previously had access to.
Advocate Ashley Edwards QC said it had the appearance of a firearm.
Mackie, from Kirkcaldy, Ellis, from Cardenden, Finlay, a prisoner and Smith, from Cardenden, all pleaded guilty to assaulting and robbing the victims.
Judge Michael O'Grady QC deferred sentence for the preparation of background reports and all four were remanded in custody.

Amnesty International supporters are zipped into some of the bags in the first few rows, alongside a banner saying #DontLetThemDrown.
Bags are lined up in rows in front of the East Sussex resort's Big Wheel close to the pier.
The protest comes ahead of an EU leaders' meeting over the crisis.
David Cameron will attend the summit in Brussels, which will discuss a response to the growing problem.
Amnesty UK director Kate Allen said: "Until now, the UK government's response has been shameful but finally they have been woken up to the need to act.
"EU governments must now urgently turn their rhetoric into action to stop more people drowning on their way to Europe."
(Close): Shares in London and other key European markets have risen on news that eurozone leaders have reached a deal on a third bailout for Greece.
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