Statistics on our Match Centre showed a total of 25 efforts at goal and Bogdan credited with 13 saves - that perfectly summed up the night.
In the end we had to wait until the 75th minute for the breakthrough as Nelsen looped home a header from Rafa van der Vaart's corner.
Gareth Bale calmly slotted home the second goal two minutes later and if there was the mildest threat of a comeback when Kevin Davies slid home for Bolton in injury time, it was soon ended by Louis Saha's cracker with the final kick of the game.
So it's Wembley next and a mouth-watering semi-final against Chelsea scheduled for Sunday, April 15.
Harry Redknapp made four changes from the team that drew 0-0 against Chelsea on Saturday.
Carlo Cudicini returned in goal, Ledley King and Ryan Nelsen paired up in central defence and Jake Livermore was back in central midfield.
Bogdan kept the tie alive with a stunning performance in the first half.
The Hungarian made five saves, including two double saves and a point-blank effort to deny Scott Parker.
The first double save came in the eighth minute when Bogdan kept out Livermore at the near post and was then on his feet again to tip over Luka Modric's follow up.
The movement of Bale, Modric, van der Vaart and Adebayor turned Bolton here, there and everywhere and they combined for van der Vaart's curler at Bogdan in the 13th minute and then a one-two between Bale and Adebayor led to the keeper blocking the Welshman's effort at the near post.
Minutes later and it was double save number two. This time Bogdan got down low to his left to touch away Adabayor's curler and was up again to block Bale's follow up.
The only lull in the game lasted until the 36th minute when Bale drilled inches wide, Bogdan then somehow turned Parker's effort away from six yards and Bale screamed another shot even closer from van der Vaart's lay-off.
The second half followed exactly the same pattern.
With just three minutes gone, van der Vaart lifted a free-kick over the wall, beating Bogdan's dive but the ball rattled the crossbar.
Ledley King was next to try his luck as he connected powerfully with van der Vaart's cross only to see his header brush the roof of the net.
There was a gasp as Bolton's first serious effort from Chris Eagles deflected off Parker and looped onto the roof of thet net on the hour - a reminder that despite our dominance, we needed that goal.
The next 10 minutes were nothing short of amazing.
Adebayor sent van der Vaart clean through only for Bogdan to save with his right foot, Kyle Walker's volley flew just wide, van der Vaart returned the compliment to set-up Adebayor but Bogdan saved again, Livermore missed the target from close range with a header, Cudicini saved from Ivan Klasnic, Jermain Defoe was denied by Bogdan with his first touch and then from the corner, the keeper was there agin to tip over Bale's header. Incredible stuff.
Just when a few doubts must have been creeping in, Nelsen struck.
And it was such a routine goal as well, van der Vaart floated a corner to the far post, Nelsen rose well and looped his header back over Bogdan into the far corner.
The relief was tangible and it was soon 2-0 as Defoe picked Bale's charge through the middle and there was no stopping him this time, a calm, side-foot finish into the corner.
Defoe went close twice before late hope for Bolton as Kevin Davies touched Klasnic's cross back across Cudicini as the game ticked into the 90th minute.
But Saha had the final word, a dream finish from 25 yards sailed into the corner to take us back to Wembley.
Spurs (4-5-1): Cudicini; Walker, King, Nelsen, Assou-Ekotto; Van der Vaart (Rose, 80), Livermore, Modric, Parker (Defoe, 71), Bale; Adebayor (Saha, 80). Unused subs: Friedel, Kaboul, Giovani, Kranjcar.
Bolton (4-3-3): Bogdan; Ricketts, Boyata (K Davies, 75), Knight, Alonso; Reo-Coker, M Davies, Pratley (Ream, 28); Miyaichi (Tuncay, 79), Klasnic, Eagles. Unused subs: Jaaskelainen, Vela, Petrov, Ngog.
Goals: Spurs - Nelsen (75), Bale (77), Saha (93). Bolton - Kevin Davies (90).
Referee: Mr H Webb.
The New Zealand defender was the unlikely hero after we'd thrown the kitchen sink at the visitors only to be kept at bay time and time again by Bogdan in the visitors goal.