Zambia striker Daka, a close-season signing from Salzburg, became only the fifth player to score four goals in a Europa League game since the competition was rebranded from the UEFA Cup, and the first since Willian Jose for Real Sociedad in 2017.
Brendan Rodgers’ Leicester moved up to second place in Group C behind leaders Legia Warsaw, a point above third-placed Spartak.
“It’s been an amazing few days for me and the team,” Daka told BT Sport. “It was a very good effort from the team. I’m just happy I was able to help the team.”
Goals from Aleksandr Sobolev and Jordan Larsson put Spartak in control, but Daka netted his first European goal for the visitors just before half-time.
Less than three minutes after the break, Kelechi Iheanacho claimed his second assist, crossing low for Daka to tap in from close range.
Leicester, and Daka, made it three goals in the space of nine minutes when the 23-year-old slid the ball under home goalkeeper Aleksandr Maksimenko.
He completed his four-goal haul with 12 minutes remaining by picking out the corner of the net after collecting James Maddison’s pass.
That goal made Daka, who also netted in Leicester’s 4-2 win over Manchester United on Saturday, the Foxes’ all-time joint-top scorer in Europe.
Sobolev gave Spartak hope in the 86th minute, but Leicester held on.