The Premier League side needed to win whilst hoping that Rubin Kazan lost in their game away to PAOK Salonika, and also needed to overturn a goal difference deficit to the Russians of five.
Spurs manager Harry Redknapp has not hidden his lack of interest in this competition this season, and he sent out a second-string team backed up by just three outfield players on the bench.
Nevertheless, at half-time it looked as if Spurs might complete a remarkable turnaround and join Stoke City as well as Champions League dropouts Manchester United and Manchester City in Friday's draw for the last 32.
In Greece, Rubin had to face playing for almost 80 minutes with 10 men after goalkeeper Sergei Ryzhikov was sent off for giving away a penalty which Vieirinha converted to give already-qualified PAOK the lead.
Meanwhile, Tottenham raced into a 3-0 interval lead against their hosts, who had lost all five previous group games.
First, a Pienaar shot from the edge of the area was deflected into the net just before the half-hour.
Then, youngster Townsend doubled the Premier League side's lead seven minutes before the interval with a brilliant curling effort from 18 yards, before a neat Defoe turn and finish from six yards then put Spurs 3-0 up right at the end of the first period.
At that point, the two-time UEFA Cup winners were just one goal away from going through, but Rubin crucially drew level in their game three minutes after the restart.
Gokdeniz Karadeniz crossed for Paraguayan international Nelson Valdez to volley home first-time and give the visitors the point they needed.
Back in Ireland, Tottenham were pushed back for most of the second-half and Carlo Cudicini should have given away a penalty and been sent-off for bringing down Ken Oman in the box.
Unfortunately for the hosts the flag had, wrongly, already been raised for offside and instead Spurs went on to make it 4-0 at the death when teenage substitute Kane struck.
Tottenham can now focus all their energies on their challenge at the top of the Premier League, although Championship club Birmingham City and Scottish giants Celtic could still progress with victories later on Thursday.
On Wednesday, Fulham were knocked out after squandering a two-goal lead to draw at home to Denmark's Odense.
Source: AFP
Source: AFP