Quincy Promes netted a brace, while Lorenzo Melgarejo, Denis Glushakov and Luiz Adriano added one apiece to propel Spartak to a thumping win.
Defender Simon Kjaer meanwhile was on target for Sevilla.
The win put Spartak level on points in Group E with Liverpool, who thrashed Maribor 7-0.
“It was difficult to face such a strong team which has won the Europa League three times, but my players did really well tonight,” said Spartak manager Massimo Carrera. “They proved that we are also capable of playing quality football.
“We looked a bit passive in the beginning of the second half, allowing Sevilla to create a set of chances. Luckily, (Alexander) Selikhov and (Dmitry) Kombarov produced key stops and it changed the game radically as we managed to take over the initiative and win.”
Spartak, spurred on by a capacity 44,000 home crowd, took an 18th-minute lead when Dutchman Promes headed the ball into the net from 10 yards off a right-wing cross by Andrei Eschenko following a rapid counter-attack.
Sevilla replied positively and Danish international Kjaer pulled them level just before the half-hour mark, sending home a rebound after a corner by Ever Banega.
As the interval approached Spartak ‘keeper Selikhov pulled off a remarkable save to parry a shot which Sevilla midfielder Michael Krohn-Dehli curled towards the top right-hand corner.
After the restart Sevilla created a host of chances, forcing Selikhov to produce a series of impressive saves to deny dangerous attempts by Wissam Ben Yedder and Krohn-Dehli.
Spartak soon regained the initiative when Paraguayan first-half substitute Melgarejo restored the Muscovites’ lead in the 58th minute by beating Sergio Rico with a low shot from the edge of the area.
Skipper Glushakov and Brazilian Adriano both increased the lead before Promes rounded off the scoring in the last minute.
“Today we made too many indivudual errors in defence and Spartak made us pay almost for every one,” Sevilla boss Eduardo Berizzo said.
“Spartak have an impressive attacking line formed of individually strong footballers, who proved their strength clearly.
“We dominated the play, while Spartak scored goals tonight. Sometimes it happens in football. But we need to withstand this blow and prepare to avenge it.”
The two teams will meet again in Seville in a fortnight.
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