After losing to AS Monaco in their opening Group E fixture, the pressure was on Mauricio Pochettino's men to get points on the board, but it took them 71 minutes to find the breakthrough in Russia.
Spurs started the match on the front foot, although they were given a warning from CSKA when Aleksandr Golovin crossed high from the byline to Mario Fernandes, but the defender squandered a scoring opportunity by heading across goal, allowing the visitors to clear.
Early in the first half, the Dutchman got on the end of Kieran Trippier's low cross, but was offside when he sent the ball over the top of the crossbar.
The summer signing was later caught offside again, and that time he managed to get the ball in the net by chipping it over the top of the keeper, but it did not count.
Dele Alli almost caused trouble for his team when he gave possession away in his own half, which prompted Lacina Traore to latch on and charge into the box, but Toby Alderweireld muscled him off the ball.
Spurs were dominant on the ball in the first half, but they failed to create many clear-cut chances, and they were dealt some nervy moments at the other end when CSKA were on the attack.
Alli's control let him down when he got on the end of Christian Eriksen's 40-yard pass, but a few minutes later, he came close to netting a 22-yard effort until the ball clattered off the bar.
Towards the end of the first half, Alderweireld had an opportunity to open the scoring when Trippier's cross met him in the 18-yard box, but his header was directed straight at the keeper.
After the break, the tempo continued at a fast pace, with both teams creating goalscoring opportunities. Son Heung-min managed to get in behind CSKA's defence, but the hosts dropped back to block his shot.
The ball rebounded back into the South Korean's path, and instead of unleashing a second attempt, he cut it back to Erik Lamela on the edge of the box, but the Argentine was crowded out.
CSKA swiftly broke away and the ball ended up at Tosic's feet in the box, but his eight-yard shot deflected over the top for a corner kick, which came to nothing.
Janssen had claims for a penalty waved away by the referee after he was clipped in the box while trying to turn and get a shot away around the hour mark.
In the final 20 minutes of the match, Lamela pushed up further forward after Janssen was substituted and he came close to breaking the deadlock from a corner, but his shot just dropped inches wide of the post.
Tottenham's pressure eventually paid off when Son got on the end of Lamela's pass and squeezed the ball past Akinfeev to give his team a much-needed goal.
The South Korean came close to a second strike minutes later when the ball dropped to him in the 18-yard box, but Georgi Schennikov lunged in with a well-timed block.
CSKA tried to force an equaliser in the closing minutes of the match, and Spurs goalkeeper Hugo Lloris almost suffered a blunder when he initially spilled Roman Eremenko's 25-yard shot, but gathered on his second attempt.
Fernandes was left with his head in his hands when his cross-shot flashed across goal with five minutes of normal time remaining.
Tottenham managed to hold on to their lead and put themselves second in the group, one point behind leaders Monaco, while Bayer Leverkusen and CSKA sit third and fourth respectively.