The supplies put India at risk of sanctions from the United States under a 2017 US law to deter countries from buying Russian military hardware.
"The first supplies have already been started," Interfax cited Shugayev as saying on November 14 at an aerospace trade show in Dubai.
He said that the first unit of an S-400 system would arrive in India by the end of this year.
The $5.5 billion deal for five long-range surface-to-air missile systems, which India says it needs to counter a threat from China, was signed in 2018.
India faces a range of financial sanctions from the United States under Countering America's Adversaries Through Sanctions Act (CAATSA), which names Russia an adversary alongside North Korea and Iran for its actions against Ukraine, interference in the US 2016 elections and help to Syria.
New Delhi said it has a strategic partnership with the United States and Russia, while Washington told India it was unlikely to get a waiver from CAATSA.
Last year, the United States imposed sanctions citing CAATSA on NATO ally Turkey to acquire S-400 missiles from Russia. The sanctions targeted the main Turkish defence procurement and development body Presidency of Defence Industries.
Washington also removed Turkey from an F-35 stealth fighter jet program, the most advanced aircraft in the US arsenal, used by NATO members and other US allies.
Russia said it had offered Turkey its help in developing advanced fighter jets, but no agreement has been reached.
"We are still at a stage of negotiations on this project," RIA news agency quoted Shugayev as saying on November 14.