The game, replayed after the collapse of Fabrice Muamba in this fixture 10 days ago, looked to be heading towards a stalemate as Adam Bogdan produced a heroic goalkeeping display to deny Spurs for 74 minutes.
But Spurs eventually broke the deadlock when Nelson scored his first goal for the club, heading in from Van der Vaart's corner.
Bale made it two in two minutes, capping a sweeping counter-attack with an assured finish into the bottom corner before Kevin Davies came off the bench to pull one back.
But substitute Saha fired in an emphatic late goal to ensure Spurs reach the last four for the 19th time.
The world of football joined together to wish Fabrice Muamba well after his collapse stunned White Hart Lane into silence 10 days ago and, ahead of kick-off, players and officials wore t-shirts with the words "Uniting for Fabrice" on what was an emotional return to Tottenham for Bolton.
Fans, too, paid their tributes with chants of "Fabrice Muamba" heard throughout.
The cardiologist who ran onto the pitch to offer his expertise to the stricken Muamba, Dr Andrew Deaner, was an honoured guest this time around - and took his place among the softer, more exclusive seats at White Hart Lane.
And Spurs fan Deaner was treated to some encouraging moments early on as Jake Livermore and then Luka Modric were both denied by Bogdan's superb double save.
Emmanuel Adebayor created the chance, cutting in from the right and threading the ball through to Livermore, whose first-time shot was blocked by Bogdan before the Hungarian goalkeeper got up quickly to brilliantly tip over Modric's drive from the edge of the box.
Spurs stamped their authority on the game early on and continued to create chances, with Adebayor and Bale both testing Bodgan, who showed why he continues to keep Jussi Jaaskelainen out of the team.
But Scott Parker should have made Tottenham's dominance count when Adebayor's superb dummy from Bale's low cross let Parker in but, from eight yards, the England captain shot straight at Bogdan.
Van der Vaart struck the bar with a beautiful free-kick, and the Dutchman should have put Spurs ahead just after the hour when he found himself clean through, but his low shot was saved by the inspired Bogdan.
Bale had a header tipped over as Spurs through everything at Bolton in increasing desperation, but they eventually broke Bolton's resistance with 16 minutes to play when Nelson headed in.
Bolton barely had time to dust themselves down before they found themselves two goals behind, with Bale finishing a slick counter-attack with a neat finish into the bottom corner.
Bolton appeared capable of scoring on the break all evening and gave Tottenham something to think about late on when Ivan Klasnic fired a low ball in from the right that Nelson could not deal with, leaving Davies the simple task of tapping in from close range.
Under Harry Redknapp, Tottenham's best form of defence has been attack and they showed this again - with Saha firing in a fine effort from the edge of the box to confirm Tottenham's trip to Wembley.