It was only the fourth time in the famous old knockout competition that a semi-final had featured the current top two in the top flight and it did not disappoint. Brazilian Willian, surprisingly selected ahead of Hazard in the starting line-up, twice put Chelsea ahead in a sizzling contest, first after four minutes with a free kick and then close to halftime from the penalty spot.
Twice Tottenham roared back with Harry Kane heading them level after 18 minutes and fellow England international Dele Alli making it 2-2 with a superb finish in the 52nd minute.
Tottenham, who are four points behind Chelsea in the Premier League after a brilliant run of seven successive wins, dominated at times but their hopes of a first FA Cup final since 1991 were sunk late on by two moments of magic.
First Hazard, who replaced Willian with half an hour remaining, buried a low shot past Hugo Lloris after 75 minutes, then Matic's 25-metre piledriver sank Spurs who have won only one of their last nine matches at Wembley and have now lost in seven successive FA Cup semi-finals.
It was a huge win for Chelsea who have wobbled lately with two league defeats and they will face either Arsenal or Manchester City in the final when a "double" could be on offer.
After a resounding ovation for former England defender and Tottenham's Under-23 coach Ugo Ehiogu who died on Friday the semi-final began at a high tempo.
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Tottenham's Dele Alli in action with Chelsea's N'Golo Kante and Cesar Azpilicueta Reuters
Chelsea, with Diego Costa and Hazard surprisingly on the bench, settled quickly and struck first.
Spurs defender Toby Alderweireld was a touch fortunate to escape a red card when his desperate tackle ended Pedro's marauding run, but greater punishment was exacted by Willian who curled a beautiful free kick past a flat-footed Lloris.
Tottenham were rattled as Chelsea swarmed around the pitch threatening a second goal but Chelsea's lead lasted only 13 minutes.
Christian Eriksen's corner was blocked but when the ball came back to the Dane he curled in a low cross and Kane, almost kneeling, feathered a delicate header beyond Thibaut Courtois for his 26th goal of the season.
Alli was becoming more involved and left David Luiz in agony after accidentally treading on the Chelsea defender's ankle.
As Tottenham assumed control Eric Dier went close to giving them the lead with a glancing header from Jan Vertonghen's teasing delivery that floated just wide.
Chelsea regained the lead out of the blue.
Eating up the ground on a forward run N'Golo Kante slipped a ball towards the overlapping Victor Moses down the right and he was sent sprawling to the turf by Son Heung-min, playing as a left wing back, who needlessly slid in.
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Chelsea's Victor Moses in action with Tottenham's Jan Vertonghen and Kieran Trippier Reuters
After a short delay referee Martin Atkinson pointed to the penalty spot and Willian dispatched it past Lloris.
Tottenham hit back again in brilliant fashion seven minutes after the break when Eriksen's teasing ball left Luiz floundering and Alli, timing his run to perfection, stabbed the ball high past Courtois.
Conte reacted by introducing Costa and Hazard with Willian making way and it was Hazard who stunned Tottenham when he was allowed to much space and he fired home through a crowd.
Matic then sent Chelsea's fans into ecstasy with an unstoppable fourth.
Willian 5,43(p), Eden Hazard 75, Nemanja Matic 80 Tottenham Hotspur 2
Harry Kane 18, Dele Alli 52