The financial details were not disclosed but media reports said Azmoun, who has signed a 3-1/2 year deal, cost 12 million euros (10.50 million pounds). Barrios joined for 4-1/2 years.
The two players were brought in shortly after Zenit sold Argentina midfielder Leandro Paredes to Paris Saint-Germain for a reported 45 million euros ($51.5 million).
The extended winter break in Russia means the transfer window in the country does not close until Feb. 22.
Zenit are top of the Russian league by one point ahead of FC Krasnodar after 17 games. Azmoun and Barrios could make their competitive debuts against Fenerbahce in the Europa League last 32 first-leg clash in Turkey on Feb. 12.
The 24-year-old Azmoun, known as the 'Iranian Messi', helped Iran reach the Asian Cup semi-finals last month in the United Arab Emirates, where they lost 3-0 to eventual runners-up Japan.
He began his career at Sepahan in Iran's top flight before moving to Rubin Kazan in 2013 and then Rostov in 2016.
He helped Rostov finish runners-up in the 2015-16 Russian league before returning to Rubin last year. In his 5-1/2 seasons in Russia, he played 154 matches and scored 42 goals.
Azmoun is approaching 50 caps for Iran, scoring 28 goals.
Barrios, 25 started out at Deportes Tolima, making his debut in February 2013 and 18 months later won the Colombia Cup. He made over 100 appearances for Tolima scoring three goals.
The midfielder joined Boca Juniors in 2016 and helped them win the Argentine title in 2016/17 and 2017/18 as well as reaching the 2018 Copa Libertadores final.
Barrios, who has joined Zenit on a 4-1/2 year deal, reached 60 appearances for the South Americans and scored one goal.
He has 17 caps, including three at the 2018 World Cup.
Last month, Zenit, bankrolled by the state-owned gas giant Gazprom, also secured a deal to sign Ukrainian defender Yaroslav Rakitsky, 29, from Shakhtar Donetsk.
(Reporting by Vladimir Soldatkin; Editing by Ken Ferris)
