"The relations are extremely difficult," Margvelashvili told reporters after sealing a landslide win at elections on yesterday.
"But we are steadfast that there will be no aggression, no destructive actions from us."
Saakashvili's key ambition of making Georgia a member of NATO and the European Union enraged Russia.
Margvelashvili pledged that furthering Georgia's integration into NATO and the EU remained the main foreign policy priority.
His first foreign trip is expected to be to Lithuania where Georgia is hoping to initial a landmark free trade agreement with the European Union at a summit in late November.
Margvelashvili said while the government of billionaire Prime Minister Bidzina Ivanishvili had managed to "lower the temperature" over the past year the fate of the two territories remains non-negotiable.