Lavrov said in a subsequent statement that both sides wish to develop a "constructive partnership" and "close cooperation."
He also says all bilateral agreements signed during the tenure of late President Saparmurat Niyazov will be honored.
Niyazov died in December, and Turkmens are due to elect his successor on February 11.
It is unclear whether Lavrov and Meredov discussed the fate of environmentalist Andrei Zatoka.
A Russian-Turkmen citizen, Zatoka was detained in Dashhowuz on December 17 and reportedly charged with illegally possessing a weapon and poisonous substances. His arrest took place four days before news of Niyazov's death was made public.
Russia's ITAR-TASS news agency on January 16 reported that Moscow is seeking access to Zatoka and intends to discuss the issue "at foreign ministers' level."