The English Premier League side, with only three outfield players on the substitutes bench, won 4-0 at Shamrock Rovers but finished third in Group A, one point behind the Russians.
Tottenham needed a win combined with a defeat for Rubin and had to make up a difference of five goals to qualify and, although it looked impossible to start with, it was on the cards at the break in the respective games.
"At half-time we knew that the other team [Rubin] were losing and thought we had a real chance," Spurs manager Harry Redknaopp told ITV4. "We knew they had gone down to 10 men."
Rubin fell behind to a Vierinha penalty in the 16th minute after their goalkeeper Sergei Ryzhikov was sent off for scything down Giorgos Georgiadis, while a second-string Tottenham side had opened up a three-goal half-time lead at Shamrock.
But Paraguay forward Nelson Haedo Valdez sscored for Rubin three minutes after the break, volleying in from the penalty spot after Gokdeniz Karadeniz had cut the ball back.
South Africa midfielder Steven Pienaar, with his first goal for Spurs, Andros Townsend and Jermain Defoe struck in a 16-minute spell before half-time to keep the Londoners in the hunt.
Substitute Harry Kane then added the fourth in stoppage-time for Tottenham, who have fielded under-strength teams in all their group games.
Shamrock's defeat in coach Michael O'Neill's last game in charge, ended a 12-match European campaign that began in mid-July and included progress to the third qualifying round of the Champions League where they lost to FC Copenhagen.
That sent them into the Europa League play-off round where they surprisingly eliminated Partizan Belgrade, only to lose all six group games.
