Barcelona had already secured top spot in Group G ahead of Tuesday's final round of matches, but Celtic needed to better Benfica's result at the Nou Camp to progress.
Gary Hooper got Celtic off to a flying start before Spartak equalised through Brazilian Ari just before the break.
The second half was a nail-biting affair for Celtic fans before a questionable barge on Georgios Samaras earned the Bhoys a penalty and Kris Commons stepped up to steer Celtic through.
"I think it is a wonderful achievement," Collins told Sky Sports. "It was a tough group and Neil Lennon and his players can be very proud of themselves - they have done Celtic proud and Scotland.
"I thought the second half was a really good performance where they really took the game to the Russians.
"There was a period where Spartak kept the ball and the Celtic fans were getting a bit anxious but the Celtic players kept their shape, kept working - none more so than Gary Hooper.
"He is a centre forward and he scored the goal but that second half he was defending and I think that is was epitomises this Celtic side.
"That is part of being a striker, it is not just about scoring goals, if you don't have the ball then you have to defend for your teammates and that is exactly what he did. Lennon has instilled in them a work rate that has taken them into the last 16 of the Champions League."
With Benfica drawing 0-0 at Barcelona, it could have gone horribly wrong for Celtic after they conceded, but Collins said they showed a lot of courage and resolve, especially in the second half
"They conceded at a bad time - it would have been a tough team talk for Lennon. Do they sit back and hope Barcelona win, or do they go out in the second half, be positive and try for another goal?" added Collins.
"I thought they came out really positive and had a couple of chances early doors. I think Celtic got a bit of fortune with the penalty - I don't think it was a penalty. If you watched it from the side, Marek Suchy has not touched Samaras and he has fallen over - it is a soft penalty."