The Armenia international, who joined Spartak from FK league rivals Krasnodar in December, struck twice in the opening 30 minutes to give the home side a comfortable lead at halftime.
He added a third midway through the second half to take his season's tally to 12 goals before Brazilian Mauricio replied for the Chechen side late in the game at Moscow's Luzhniki stadium.
"It's the first hat-trick of my professional career, coming in my first game for Spartak in front of our fans," Movsisyan told reporters with the game ball in his hands. "No doubt, this is a special day for me as I'm taking this ball home with me."
Elsewhere, second-placed Anzhi Makhachkala suffered a shock 2-0 defeat at bottom-of-the-table Mordovia Saransk to drop five points behind league leaders CSKA Moscow, who beat Krylya Sovietov Samara 2-0 on Saturday.
"Obviously, we're very disappointed. We controlled the game but just couldn't find the target," said Anzhi coach Guus Hiddink, whose team were held to a 0-0 draw by Newcastle United in a Europa League last 16 tie on Thursday.
"I don't think we underestimated our opponents today. I think we just lost concentration at certain moments and it cost us the game as both of their goals came after corners."
Champions Zenit St Petersburg also lost, going down to Rubin Kazan 1-0 after a 31st-minute goal by Venezuela striker Salomon Rondon. The game was played in the Chechen capital Grozny because Kazan's pitch has been in poor condition.
"We had a lot of chances but came away empty-handed and they scored on a counter-attack," conceded Zenit coach Luciano Spalletti, whose team stayed third, eight points off the pace.