The 'impossible job' proved exactly that as our Europa League campaign ended despite a 4-0 win in Dublin on Thursday night.
We needed to win by five goals and hope PAOK beat Rubin Kazan to take the unlikely step into the knockout stages.
But it all looked on at half-time as we led 3-0 at the Tallaght Stadium and PAOK were 1-0 up in Greece, with Rubin's goalkeeper sent off and the Russians down to 10 men.
However, Rubin equalised in the second half, we could only add one more goal and a draw in Salonika was enough for PAOK and Rubin to both go through.
It would have been all so different had we even drawn in Matchday Five against PAOK at the Lane and the poor start in that game - 2-0 down inside 20 minutes - ultimately cost us.
Harry Redknapp talked about a tough task in Tallaght and that's how it turned out in a hard-fought opening half-hour.
Niko Kranjcar returned after injury in central midfield in a strong team including the likes of Younes Kaboul, Benoit Assou-Ekotto, Sandro and Jermain Defoe.
Rovers worked their socks off in the opening period and there were few chances at either end.
Defoe let fly a couple of times from 20 yards and Danny Rose's cross struck the crossbar, but we clicked into gear in the 29th minute.
Picking the ball up on the right corner of the box, Steven Pienaar struck low across Rovers' keeper Richard Brush and into the far corner.
Defoe almost immediately slid in the second from Giovani's cross and Kranjcar tested the keeper before we made it 2-0 on 38 minutes.
Danny Rose played into Defoe and he set-up Andros Townsend to curl a delightful effort into the top corner.
Two became three on the stroke of half-time as Townsend skipped past Pat Sullivan and crossed for Defoe to control and fire home from close range.
The home side got back into shape for the start of the second half and instead of us being able to add a fourth, Rovers went close through Billy Dennehy and Stephen O'Donnell.
We were soon back on the offensive, however, as Sandro struck the post and both Pienaar and Townsend saw efforts blocked.
Chris Turner headed just wide for Rovers and we then poured forward for the next 10 minutes. Harry Kane replaced Defoe and immediately tested Brush, Kranjcar curled a fraction wide and our second substitute, Yago Falque, struck the post from 20 yards.
Carlo Cudicini denied Sullivan a spectacular consolation with a flying save and Kane had the final word with his first senior Spurs goal, turning home from Townsend's nod down in the final minute.
It wasn't quite enough - but if we'd managed that draw against PAOK at home, we'd be in the last 32.
Shamrock Rovers (4-2-3-1): Brush; Sullivan, Oman, Murray, Stevens; Rice (O'Donnell, 46), Turner; Dennehy, Finn (Twigg, 57), Paterson; Sheppard (Kilduff, 75). Unused subs: Thompson, McCabe, O'Neill, Ricketts.
Spurs (4-4-1-1): Cudicini; Assou-Ekotto (Falque, 84), Kaboul, Livermore, Rose; Townsend, Kranjcar, Sandro, Pienaar; Giovani; Defoe (Kane, 76). Unused subs: Gomes, Carroll.
Goals: Spurs - Pienaar 29, Townsend 38, Defoe 45, Kane, 89.
Referee: Mr S Studer.