Harry Redknapp's side started the night knowing a point was enough to secure their passage from Group D into Friday's draw for the last 32.
But his players failed to read the script as they were slow out of the blocks at White Hart Lane and trailed 2-0 at the interval.
However, a second-half fightback secured the stalemate they needed to progress to the relief of the manager, who had made six changes to the side that secured a point against Manchester United at the weekend.
Chris Gunter, Gilberto and Gareth Bale started in defence, Tom Huddlestone and Jamie O'Hara were drafted into midfield, while Fraizer Campbell was employed as a lone striker in the absence through illness of Darren Bent and the cup-tied Roman Pavlyuchenko.
The wholesale changes appeared to have a detrimental effect on the home side as they began slowly and allowed Spartak to dominate the early exchanges.
Brazilian goalkeeper Heurelho Gomes was forced into a couple of early saves from Vladislav Rizhkov and giant centre forward Artem Dzyuba.
The error-prone keeper showed his new found confidence gained on the back of recent impressive displays when he tipped a Rizhkov shot around his post on 15 minutes.
But there was little the keeper could do when a comedy of errors gifted the visitors an opening goal on 23 minutes.
Gilberto hesitated and allowed a pass from Bale to roll past him towards Gomes but, to the watching full-back's horror, Dzyuba nipped in and tucked the ball between the helpless keeper's legs.
Spurs began to offer more of an attacking threat, but White Hart Lane was stunned on 33 minutes when Dzyuba doubled his side's advantage.
He nutmegged Didier Zokora 30 yards from goal before rounding the advancing Gomes and sliding the ball into the empty net.
Dzyuba could have had a first-half hat-trick but Gomes parried his deflected effort just before half-time.
Campbell came close to halving the deficit as Spurs started the second half at a high tempo, but the on-loan striker looked on in disbelief as Stipe Pletikosa clawed away his close-range header from O'Hara's cross.
The home side continued to ask questions and Luka Modric should have scored in the 58th minute but he dwelled on the ball in the penalty area and his shot was compromised by the arrival of two defenders.
Half-time substitute Aaron Lennon looked lively down the right flank and it was no surprise when he set-up Modric for Spurs' first of the night on 68 minutes.
The diminutive winger danced past two defenders before his pull back was confidently dispatched on the volley by Modric for his first goal for the club since arriving in the summer.
Spurs were back on level terms in the 74th minute when Bale's pinpoint cross found the on-rushing Huddlestone and his powerful 12-yard header left the keeper with no chance.
Campbell wasted two gilt-edged chances to win the game in the final ten minutes and saw a penalty appeal rejected, but in the end the north London secured the point they needed to ensure their name will be in the hat for Friday's draw as group runners-up.