Gas deliveries started shortly after 10 am local time, said Vladimir
Ivanov, spokesman for the emergency situations ministry in the Russian border province of North Ossetia.
Residents in the impoverished former Soviet republic have been suffering severe energy shortages since the explosions on January 22 wrecked the pipeline.
The pipeline carried nearly all of Georgia's natural gas supplies from Russia.
Electricity supplies were also disrupted when a separate explosion targeted a power line from Russia across the Caucasus into Georgia.
Georgian Prime Minister Zurab Nogaideli confirmed that gas supplies were being restored but said some people would be without full power up to another week.
The Tbilgaz company said part of Tbilisi should start receiving gas on Sunday and the rest of the capital on Tuesday or Wednesday.
Repairs to the Russian pipeline were completed on Saturday following what officials said were repeated difficulties in carrying out the work high in the snow-covered Caucasus mountains.
Russian officials said "terrorists" were behind the blasts that damaged the gas and electricity condiuts.
No further clues have been offered as to the alleged attackers' identity.
Georgia's government suggested that the Russian authorities may have orchestrated the blasts to apply economic pressure on its pro-Western neighbour in the middle of a harsh winter.
The Russian government has denied the charge.
In the wake of the crisis, Georgian President Mikheil Saakashvili vowed to end their country's almost total dependence on Russian energy sources.
"The Georgian authorities will do everything so that our people are never again put in such a situation," Nogaideli told reporters on Sunday."
Everyone saw clearly that Georgia cannot be broken."
Row drags on
The end of repair work Saturday has not put an end to the gas dispute.
Tbilisi Mayor Gygy Ugulva announced late Saturday that gas supplies to the Russian embassy and a subsidiary of Russian giant Gazprom, Gazexport, had been cut off.
Mr Ugulva accused the two of being "organisations that have been directly taking part in Georgia's energy blockade".
But gas supplies to the embassy were restored late Sunday after the Russian foreign ministry threatened tit-for-tat action.
President Saakashvili declared last week that "Georgia will never be brought to its knees."
Tensions between Georgia and Russia have grown since the 2003 "rose revolution" that brought Saakashvili to power, promising a new pro-Western course and renewed efforts to join the North Atlantic Treaty Organisation (NATO).
Russia accuses Georgia of allowing Chechen rebels to take refuge on its territory, while Tbilisi has for more than a decade accused Moscow of funding and arming separatist rebels who control two key Georgian provinces – South Ossetia and Abkhazia.