The Hoops held on for a 1-1 draw in the third qualifying round first leg despite being overrun in the first 20 minutes.
Paul Hartley’s 21st-minute header put the visitors ahead before Roman Pavlyuchenko headed the equaliser three minutes before half-time.
But Strachan hit out at the surface in the Luzhniki Stadium. “They really watered the pitch and made it slicker,” he claimed. “We weren’t allowed to practice on that.
“They put a bit of water on it this morning but nothing like that. So the speed of the game caught us out for the first 20 minutes.
“The players couldn’t really even pass the ball to each other, they were sliding about and that wasn’t what we practiced on.
“You didn’t have a clue what was going on. When we were practicing the other day, the ball was holding up.
“It wasn’t holding up tonight, it was going straight through.”
Strachan hailed his side for adjusting to the pitch against the Russians.
“They are a right good side,” he said. “They have won nine out of 12 matches on that pitch this season.
“They know how to play on it, they know every tweak, every spin, when the ball goes long, when it kicks up.
“So playing a good side on a very unusual pitch, to handle that was fantastic.’’
Summer signing Scott Brown had a terrific chance to hit a late winner for Celtic, but the Scotland midfielder failed to convert from close range, while Spartak hit the bar in the last minute of the third qualifying round tie.
The home side looked inspired from the first whistle, forcing a startled looking Celtic side back on to the back foot.
Pavlyuchenko soon had the ball in the net from six yards after Brazilian star Welliton left Kennedy for dead on the left.
Pavlyuchenko produced a neat finish, hooking the ball in after rushing ahead of the pass, but the strike was disallowed for offside.
A clever back-heel from Welliton across the face of the Celtic goal had the visiting defence in trouble again and there seemed no respite for the Scots side.
However, the tie was turned on its head in the 21st minute when the Scottish champions took the lead with their first effort on goal.
McDonald looked suspiciously offside when he collected a Shunsuke Nakamura pass down the right — but the little striker crossed a delightful ball to Hartley, who put his header from 10 yards past Stipe Pletikosa and into the corner of the net.
But three minutes before the break the home side were level.
Roman Shishkin’s cross from the right was met by Pavlyuchenko and although his close-range header was parried by Mark Brown, the ball escaped over the line.
Pletikosa, Shishkin, Kovac, Stranzl, Soava, Mozart, Torbinsky, Bystrov (Kalinichenko 76), Titov, Pavluchenko, Welliton.
Mark Brown, Wilson, McManus, Kennedy, Naylor, Hartley, Scott Brown, Nakamura, Donati (Sno 74), McDonald (Caldwell 79), Vennegoor of Hesselink (McGeady 82).
Herbert Fandel (Germany).