"The transit of Russian gas to Europe via Ukraine, and shipment of gas to Ukrainian customers, were started in accordance with the agreements," Gazprom said in a statement on Tuesday.
Valentin Zemlyansky, a spokesman for Naftogaz, said that the gas was flowing into Ukraine's pipeline network via all points of entry from Russia.
The dispute had affected millions of people in eastern and central Europe and led to the shut-down of many schools and factories.
The European Union's Czech presidency urged caution, saying it would take several days to be sure of an end to the crisis.
|Viktor Yushchenko, Ukraine's president, was said to be 'devastated' by the deal [AFP]
Martin Riman, the Czech Republic's industry minister, said: "We will be fully satisfied after three or four days of supplies to Europe."
Barroso criticised Russia and Ukraine over their handling of the gas crisis.
He said: "I was very disappointed in these days about the way the leadership in these two countries negotiated."
The crisis betweern the two countries erupted on January 1 when Russia cut gas to Ukraine's domestic market over unpaid debts and demands for a higher price in 2009.
It escalated on January 7 when all supplies for Europe transiting through Ukraine were halted, with Moscow accusing Kiev of stealing gas.
Ukraine vehemently denied the allegation.
Andris Piebalgs, the EU energy commissioner, backed Ukraine on Tuesday, saying there was no evidence to support Moscow's claims that Kiev had stolen Russian gas bound for Europe.
"I have no evidence that Ukraine took gas without permission," Piebalgs said during a joint news conference in Kiev with Yulia Tymoshenko, Ukraine's prime minister.
As details emerged about Monday's agreement, it appeared that Ukraine had lost ground to Russia by agreeing to a significant rise in the price it would pay Moscow for gas, at least in the short term.
The office of Viktor Yushchenko, the Ukrainian president, felt "devastated" by the agreement brokered by Tymoshenko, his political rival, but would not attempt to block it, said Oleksandr Shlapak, the deputy head of Yushchenko's administration.
Gazprom said in a statement that in the first quarter of 2009, Ukraine would pay Russia $360 per 1,000 cubic metres of natural gas and the price would change quarterly.
That rate is double the $179.50 Ukraine paid last year under terms far better than the market prices paid by EU countries for Russian gas.
Europe currently pays around $450 for the same quantity of Russian gas, but the price is expected to fall dramatically later this year due to the global economic slowdown and plunging oil prices.
Tymoshenko said in Kiev that the price Ukraine would pay over the course of 2009 would average out to $228.80.
The average gas price for European customers in 2009 would be $280, said Alexander Medvedev, Gazprom's deputy chief executive.
Tymoshenko called Monday's agreement "historic" and said it would prevent future year-end gas crises, while Vladimir Putin, her Russian counterpart, expressed hope it "will allow us to build long-term, reliable and stable relations".
Under the terms of the deal, Ukraine would keep the fees it charges Russia for the transit of gas across its territory frozen at the 2008 level this year, after which they could be raised.