Willian's fifth-minute free kick handed the Blues the lead in the London derby clash before Harry Kane superbly headed in Christian Eriksen's cross to restore parity 13 minutes later.
But Willian's penalty just before the break - awarded for Heung-min Son's foul on Victor Moses in the area - gave Chelsea the lead once more before Spurs responded in fine fashion through Dele Alli, who latched onto another brilliant Eriksen cross to make it 2-2 in the 52nd minute.
With the score level at 2-2 and heading for extra-time, Conte made a host of changes which paid off when substitute Hazard coolly drove in a low shot in the 75th minute that took Chelsea 3-2 ahead.
Nemanja Matic's stunning long-range shot moments later then ensured the Blues will return to Wembley for the final against either Arsenal or Manchester City.
Despite the importance of the tie and the fact it was a London derby, thoughts before the game turned away from the pitch and to Spurs' Under-23s coach Ugo Ehiogu.
Both Tottenham and Chelsea fans stood united on Saturday as they carried out a minute's applause for the 44-year-old, who passed away on Friday after suffering a cardiac arrest at the club's training ground a day earlier.
With both teams observing the minute's applause and wearing black armbands, the tie got underway with a much-changed Chelsea side on the pitch.
Conte demonstrated to his players he isn't afraid to make big changes after watching his Chelsea side put in a lacklustre performance against Manchester United six days earlier to lose 2-0 at Old Trafford - a result that saw Spurs move to within four points of the leaders in the race for the Premier League title.
With Thibaut Courtois returning from an ankle injury alongside Marcos Alonso, who did not play against United due to a virus, Conte decided to replace Gary Cahill - sidelined with gastroenteritis - with Nathan Ake.
The full-back had made just two starts - both in this competition - since returning from his loan spell at Bournemouth early, but despite his lack of game time and, indeed, the availability of captain John Terry and Kurt Zouma, Conte insisted the 22-year-old was ready to start against Spurs.
And Ake certainly showed his composure early on when, after just three minutes, he won a tackle with Kane which allowed Chelsea to launch a swift counter attack that ended in Toby Alderweireld bringing down Pedro and being booked in the process.
Willian - starting for Hazard - whipped in a superb effort that curled in past Hugo Lloris to hand the Blues the lead within five minutes.
Conte was left ecstatic on the touchline as he looked on at his players and, moments later, he was then looking up at the sky as a plane circled over Wembley with a banner reading his name.
It was received warmly by the Chelsea fans, who have grown to adore Conte this season, before they were left applauding when another one of the Italian's changes, Michy Batshuayi, forced a save from Lloris with his header in the 14th minute.
The last time the striker started for Chelsea was in late January in the 4-0 win over Brentford in the FA Cup, but here he looked sharp in place of the misfiring Diego Costa.
However, Spurs - who Tottenham boss Mauricio Pochettino had insisted before the game were underdogs to the 'favourites' Chelsea - reacted quickly to Batshuayi's close chance as they took less than three minutes to restore parity through the in-form Kane.
After winning a corner and keeping it alive, Eriksen sent in a ball that Kane dived to head in past Ake and Courtois.
It was a deserved equaliser from a side who came into the semi-final on a high following seven straight Premier League wins and they then threatened to take the lead in the 35th minute through Eric Dier, whose header flew wide.
Chelsea were certainly finding opportunities in front of goal hard to come by until four minutes before the break when Son made contact with Moses in the box to give away a penalty.
Willian coolly converted the resulting spot-kick to make it 2-1 to Chelsea before the Brazilian and, indeed, the rest of his teammates were left complaining to referee Martin Atkinson for Lloris' apparant handball outside the area.
Despite the fact the Frenchman had touched the ball outside the box as he tried to claim it, no action was taken against the Spurs goalkeeper - playing in his first FA Cup tie since 2014 - which proved a huge let-off for Pochettino as his side were dominating the tie.
The north Londoners kept up the pressure on Chelsea's defence after the re-start and they duly were rewarded in the 52nd minute when Eriksen's superb cross found Alli, who was missed by both Cesar Azpilicueta and David Luiz, and left free to fire in the equaliser.
The goal sparked Spurs into further action but when Alli was seemingly brought down by Ake in the box, the 21-year-old was left incensed when no penalty was given.
Conte reacted by taking off Batshuayi and goalscorer Willian for Costa and Hazard respectively and then bringing on Cesc Fabregas later in the second half.
The changes proved crucial as Fabregas' corner kick in the 75th minute was half-cleared by the Spurs defence and instead fell to Hazard whose resulting low shot deflected off Alderweireld and into the goal.
But there was more to come from the Belgian and his fellow sub Fabregas as the pair then combined to tee up Matic, whose long-range effort struck the bar and into the net.
Spurs tried to respond but there was little time left for Pochettino's men to score two more and Chelsea held on to book a date with either Arsenal or Manchester City, who play their semi-final on Sunday, in the FA Cup final next month.