Lavrov also told state news agency RIA Novosti that Russia's objectives will expand still further if Ukraine's allies keep supplying it with long-range weapons such as the US-made High Mobility Artillery Rocket Systems (HIMARS).
His comments, the clearest acknowledgement yet that Russia's war goals have expanded over the five months of war, came after the US said it saw signs Russia was preparing to formally annex territory it has seized in its neighbour.
Ukrainian Foreign Minister Dmytro Kuleba retorted that Russia rejected diplomacy and wanted "blood, not talks".
European Commission President Ursula von der Leyen meanwhile accused Russia of "blackmailing" the European Union over energy as she unveiled a plan to slash gas demand in the bloc ahead of a feared cut-off of deliveries by Russia as winter approaches.
Russian President Vladimir Putin had earlier warned that gas supplies sent to Europe via the huge Nord Stream 1 pipeline, which has been closed for 10 days for maintenance, were at risk of being reduced further.
Lavrov is the most senior figure to speak openly of Russia's war goals in territorial terms, nearly five months after Putin launched his February 24 invasion with a denial that Russia intended to occupy its neighbour.
Then, Putin said his aim was to demilitarise and "denazify" Ukraine - a statement dismissed by Ukraine and its allies as a pretext for an imperial-style war of expansion.
Lavrov told RIA Novosti geographical realities had changed since Russian and Ukrainian negotiators held peace talks in Turkey in late March that failed to produce any breakthrough.
At that time, he said, the focus was on the Donetsk and Luhansk People's Republics (DPR and LPR), self-styled Russian-backed breakaway entities in eastern Ukraine from which Russia has said it aims to drive out Ukrainian government forces.
"Now the geography is different, it's far from being just the DPR and LPR, it's also Kherson and Zaporizhzhia regions and a number of other territories," he said, referring to territories well beyond the Donbas that Russian forces have wholly or partly seized.
"This process is continuing logically and persistently," Lavrov said, adding that Russia might need to push even deeper.
After being beaten back in an initial attempt to take the capital Kyiv, Russia's defence ministry said on March 25 that the first phase of its "special military operation" was complete and it would now focus on "achieving the main goal, the liberation of Donbas".
Nearly four months later, it has taken Luhansk, one of two regions that make up the Donbas, but remains far from capturing all of the other, Donetsk.
In the past few weeks it has ramped up missile strikes on cities across Ukraine.
On Wednesday, the Ukrainian military reported heavy and sometimes fatal Russian shelling amid what they said were largely failed attempts by Russian ground forces to advance.
Citing US intelligence, White House national security spokesman John Kirby had earlier accused Russia of laying the groundwork to annex Ukrainian territory it has seized since the start of the war, an assertion the Russian embassy in Washington DC said mischaracterised what Russia was trying to do.
Russia's invasion has killed thousands, displaced millions and flattened cities, particularly in Russian-speaking areas in the east and southeast of Ukraine.
It has also raised global energy and food prices and raised fears of famine in poorer countries as Ukraine and Russia are both major grain producers.
In Washington DC, US Defense Secretary Lloyd Austin said the United States will send four more HIMARS artillery systems to Ukraine, in its latest military help for the country.
In Brussels, a meeting of EU diplomats agreed a new round of sanctions against Russia that are due to target gold as well as more individuals and entities, one person involved in the talks told Reuters.
With uncertainty swirling over the planned restart on Thursday of Nord Stream 1, the EU proposed its 27 member countries cut gas demand by 15 per cent from now until the northern hemisphere spring.
