The Scottish champions' triumph came as group rivals Benfica could manage only a 0-0 draw against Barcelona in the Nou Camp, leaving Celtic two points ahead in second place in the qualifying group.
Their decisive goal came in the closing stages when Giorgios Samaras won a soft penalty and Kris Commons smashed the spot kick home off the crossbar.
Neil Lennon was unable to watch the moment. He had previously said victory could be the highlight of his managerial career, and relied on the roars from the crowd to tell him the good news.
Watching from the stands again were Rod Stewart, who famously burst into tears when Celtic defeated Barcelona last month, and Celtic legend Kenny Dalglish. Former Celtic favourite Aiden McGeady appeared on his old turf as a second-half substitute.
Gary Hooper had given Celtic the lead in the first half, but Ari equalised for Spartak, leaving a packed Parkead to endure a nervous second half.
The fans - and Lennon - had more than half an eye on the result in the Nou Camp, where Barcelona fielded an under-strength team and brought on talisman Lionel Messi only after the break. But it finished 0-0.