Taurasi, the MVP of last season's Final Four in Salamanca, poured in 20 of her game-high 24 points in the first half as Spartak rolled to a 94-51 triumph.
Spartak opened up a 9-2 lead midway through the first quarter with Taurasi scoring her team's first five points, but the visitors from Hungary clawed their way back to trail 11-10.
After Zuzana Horvath made two free-throws, Olena Zherzherunova cut the deficit to one with a lay-up.
Taurasi sank two from the charity stripe and Ilona Korstin finished a fast break to give the hosts a 15-10 advantage by the end of the frame.
When play resumed, the home side began to pad their lead.
After Vaida Sipaviciute made one of two free-throws to cut the Spartak advantage to 26-18, Pokey Chatman's team put the result beyond doubt.
Anete Jekabsone-Zogota drilled a jump shot and Korstin raced in for another lay-up at the front of an 18-0 scoring rampage that Szeviép never recovered from.
On top 44-20 at half-time, Chatman's squad returned to the floor and the American coach spread out the minutes, with Serbia international Sonja Petrovic scoring nine points and Natalia Vieru contributing eight along with five rebounds in 12 minutes.
Spartak improved to 6-0 with the victory while Szeviép slumped to 1-5.
Taurasi needed just 21 minutes to inflict her damage on the Hungarian team.
The Russian club out-rebounded Szeviép 34-18 and also forced the visitors to turn the ball over 25 times..
While Spartak were seven of 10 from long range, Szeviép were just one of 10.
"I am happy that my team are continuing to play well especially on the defensive end. I am especially happy that are continuing to build momentum, something especially important today since our opponents had just won their last game."
Sonja Petrovic, Spartak Moscow Region Vidnoje:
"At the beginning we were not as serious as we should be and only after we did [become serious] were we able to take over. I am happy to be helping my team in this effort of ours to finish first in the EuroLeague's most difficult group."
Tambs Benze, Szeviep Szeged coach:
"Congratulations to Spartak. Even though they are a much better team than us my players did not fight as much as they could had; if we were more serious in the second quarter the score might had been different."