The Scottish champions booked their place in the lucrative group stages, securing a £10million jackpot, following a breathtaking encounter at Celtic Park, with the hosts winning a tense penalty shoot-out 4-3.
The second leg, third round qualifier remained tied after the regulation 90 minutes after Scott McDonald's opener was cancelled out by Roman Pavluchenko. The Russian striker and Jan Vennegoor of Hesselink had also missed spot-kicks, while Martin Stranzl saw red for Spartak in an action-packed encounter.
Substitute Maxim Kalinichenko 's penalty miss during the shoot-out proved to be decisive after Shunsuke Nakamura and Egor Titov had both failed to hit the target.
"I'm sure all you press guys enjoyed it as a game and it's one that you will remember too.
"It was a game that had everything - a sending-off, silly football, goals, penalties.
Only afterwards did Strachan reveal his woeful record as manager facing a penalty shoot-out. Asked whether it was his first success, he admitted: "I don't think I've won one before.
"I was worried when it went to penalties because my record as a manager is poor but I went around everyone convincing them everything would be okay."
Boruc and team-mate Lee Naylor were involved in an altercation shortly before half-time but Strachan added: "I had forgotten all about that, we had too much else to talk about at half-time."
Clearly relieved to have secured their safe passage, the Celtic boss said: "I'm fine just now but I hope the telly is good about 4am. I'm going home to chill out now. Whatever happens tomorrow is fine."