A 4-0 win was not enough to see Spurs through to the last 32 as Harry Redknapp's European campaign ended in disappointment and anger in Dublin.
Goals from Steven Pienaar, Andros Townsend, Jermain Defoe and Harry Kane gave Tottenham a comprehensive victory over the Irish part-timers, but Rubin Kazan's 1-1 draw at PAOK Salonika meant the favourites for Group A crashed out.
The match took an ugly turn with 20 minutes to go when Redknapp reacted angrily to barracking from the Shamrock fans by appearing to aim a defiant clenched-fist gesture towards them from the edge of his technical area.
Despite being overwhelming favourites to top Group A when the draw was made, defeats against Rubin Kazan and PAOK Salonika with weakened sides earlier in the competition had cost Tottenham dearly and Redknapp had already written off his team's chances of qualifying before kick-off tonight.
The Londoners went in to the match tonight knowing they had to beat Shamrock and hope PAOK beat Rubin while the Spurs also had to overturn a five-goal difference the Russians held over them.
A Tottenham team containing nine changes started the half rustily, and Shamrock looked equal to their challenge early on.
Ken Oman fired wide for Rovers and a mix-up between Younes Kaboul and Carlo Cudicini almost allowed the Irish champions in.
Richard Brush was barely tested until Danny Rose's miscued shot flew goalwards and the Rovers goalkeeper back-tracked to tip on to the crossbar with 25 minutes gone.
The move seemed to spark Tottenham in to life and they started to play with more confidence.
News had filtered through that Rubin were down to 10 and 1-0 down and Spurs upped the ante, taking the lead through Pienaar just before the half hour.
The South African was given space in the box and fired into the far corner with a deflected effort from 18 yards.
Their tails up, the away side looked for a quick second and they almost got it when Giovani Dos Santos broke with pace and crossed to Defoe, but the England man arrived at the back post a millisecond too late.
Brush showed good reflexes to tip Pienaar's ambitious lob wide before Townsend, given acres of space in the box, took a touch and curled a sweet shot into the Shamrock goal to make it 2-0.
Brush came under further attack towards the end of the first half, saving well from Defoe, before the former West Ham hitman made it 3-0.
Townsend raced down the left and laid the ball off for Defoe, who spun on his heels and calmly slotted home in the final minute of the half.
Michael O'Neill, taking charge of Shamrock tonight for the last time, brought on Stephen O'Donnell for Stephen Rice at half-time and his team started the second half with much more purpose.
Pat Sullivan broke in to the box but saw his powerful half-volley blocked before Karl Sheppard raced through on to a ball over the top but went down under pressure from Jake Livermore.
The home side appealed for a penalty but referee Stephan Studer waved play on.
O'Donnell blasted a fierce half-volley just wide soon after as Rovers kept up the pressure.
Spurs were struggling to revive the flowing attacking play they had displayed at the end of the first half and their hopes of qualification were dealt a huge blow when Rubin equalised in Thessaloniki, meaning a Tottenham victory by any margin here would still see them go out.
A brilliant clip over the top from Benoit Assou-Ekotto played Sandro in, but the Brazilian's lob bounced back off the post and off Brush for a corner.
Sheppard raced on to a through ball and was hauled down by Cudicini, but the Italian's blushes were saved when the linesman correctly judged that the striker was offside.
Shamrock thought they should have had a penalty when Sheppard went down under a tackle from Kaboul in the box but the referee gave nothing.
With 20 minutes left, an irate Redknapp appeared to flick a defiant gesture towards the away fans who had chanted critical songs about him almost constantly throughout the match.
Shamrock looked the more likely team to score as the match drew to a close, but Kane, who had come on for Defoe moments earlier, broke free and wriggled past two markers before firing at Brush when Townsend was free at the back post.
Assou-Ekotto came off for Yago Falque with seven minutes left before Kranjcar curled a dipping shot just over the bar.
Falque found space and hammered a peach of a shot that beat Brush but slammed in to a post and out to safety.
Kane got on the end of Townsend's assist to tap home in the last minute, but it proved worthless as PAOK's failure to beat Rubin meant Spurs were out.