The 27-year-old was forced off in the 24th minute after going down with the injury, with Denis Cheryshev taking his place.
Russia were 1-0 up at the time of Dzagoev's injury after Yuri Gazinsky headed the hosts in front after 12 minutes.
And Dzagoev's replacement, Cheryshev, was able to make his presence felt as he scored twice to help Russia down Saudi Arabia 5-0.
Villarreal man Cheryshev blasted home Russia's second on the 43rd-minute mark before scoring a beautiful goal with the outside of his left foot to make it 4-0 in the second half.
Artem Dzyuba and Aleksandr Golovin completed the scoring in what was a dream start to the tournament for Stanislav Cherchesov's men.
Shortly after limping off, Russia released a statement confirming Dzagoev will now have a scan which could rule him out for the rest of the competition.
"Alan Dzagoev damaged the muscles in the back of the thigh," Russia said in a brief statement.
"The terms of treatment will be determined by the results of the upcoming MRI [scan]."
When healthy, CSKA Moscow man Dzagoev is an important part of Russia's efforts on the international stage. He was a standout performer during Euro 2012 qualifying, scoring four times to help Russia make the tournament.
Then at Euro 2012, Dzagoev scored three times in as many games, which was good enough to secure him joint-top scorer in the competition, alongside the likes of Cristiano Ronaldo, Fernando Torres, Mario Balotelli, Mario Gomez and Mario Mandzukic.
However, injury has plagued Dzagoev the last two international tournaments for his country. He had to pull out of Euro 2016 with a broken foot after picking up the injury playing for CSKA Moscow.
Dzagoev also missed out on the 2017 Confederations Cup, also in Russia, due to injury.
It was Dzagoev's 58th cap for Russia against Saudi Arabia. He has scored nine times with the national team so far.