Harry Kane scored the only goal of the game from the penalty spot, although it took a telling intervention from VAR to award the spot kick after the official had initially ruled Kane offside before the foul.
Chelsea pushed for an equaliser during a largely one-sided second half, but Tottenham held out against their London rivals and will now travel to Stamford Bridge for the second leg later this month with a narrow advantage.
Both managers named close to full-strength sides for the derby, and the match got off to a quick start with Ross Barkley firing wide of the near post and Kane sending a brilliant bicycle kick into the arms of Kepa Arrizabalaga inside the opening five minutes.
Callum Hudson-Odoi - starting a second successive match amid intense speculation over his future - then drew a routine stop from Paulo Gazzaniga from inside the area, although the majority of Chelsea's efforts during the opening 45 minutes came from range.
Gazzaniga made easy stops to deny both Eden Hazard and N'Golo Kante from such efforts, as Chelsea boosted their shot count without really threatening to make a breakthrough until the final five minutes of the first half.
The major talking point of the opening 45 minutes came when Kane latched on to a pass over the top and beat Kepa to the ball before being felled by the Chelsea keeper inside the area.
There was little doubt about the foul, but the penalty was initially turned down due to the offside flag being raised against Kane. However, a VAR review showed that the Spurs striker was onside and referee Michael Oliver therefore pointed to the spot - much to the dismay of the Chelsea players.
Kane stepped up himself and tucked his penalty into the bottom corner, giving Spurs the lead and in the process becoming the first player in the club's history to score 20 or more goals in five consecutive seasons.
It took until the 40th minute for Chelsea to offer up any serious response to falling behind, but they came within the width of the post of levelling things up when Kante darted across his marker and steered his volley against the outside of the upright.
Hudson-Odoi then flashed another effort off target before the visitors hit the woodwork for a second time at the end of the first half - although on this occasion they needed a sizeable deflection to strike the frame of the goal.
The lively Hudson-Odoi was again involved when his cross struck Danny Rose and threatened to loop over Gazzaniga, who reacted well to claw it against the inside of the post before Toby Alderweireld hacked it clear from his own six-yard box.
It proved to be the final action of the first half, but Chelsea were on top for much of the second half too as they pushed for an equaliser to take back to Stamford Bridge later this month.
Spurs were still the first to come close after the interval when Kane sent a swerving piledriver towards goal from 25 yards out, forcing Kepa into a smart one-handed save to his right.
Chelsea quickly responded as Gazzaniga was called in action to deny Hazard and Kante in quick succession, but the Spurs keeper was as much a spectator as the noisy Wembley crowd when the visitors spurned their best chance shortly before the hour mark.
Barkley flicked a header from Hazard's corner towards the back post where Andreas Christensen was waiting unmarked, but he went for the shot with his wrong foot and fired wide when the goal was gaping.
It proved to be the visitors' last big chance of the contest as Spurs held out against some relentless pressure in the final half-hour, handing them the slight advantage ahead of the reverse fixture at Stamford Bridge in 16 days' time.
The result means that Spurs have now won three games in a row against Chelsea for the first time since a run of five ended in 1963, while the Blues become only the second team to lose twice against the same opposition at Wembley in a single season.
More importantly, though, it takes Mauricio Pochettino a small step closer to his first piece of silverware both in charge of Spurs and in his managerial career, with his side now just 90 minutes away from a final which will also take place at Wembley.
TOTTENHAM (4-4-2): Gazzaniga; Trippier, Alderweireld, Sanchez, Rose; Winks (Skipp 90'), Sissoko, Eriksen (Llorente 91'), Dele; Son (Lamela 79'), Kane
CHELSEA (4-3-3): Kepa; Azpilicueta, Christensen, Rudiger, Alonso; Kante, Jorginho, Barkley (Kovacic 75'); Hudson-Odoi (Giroud 79'), Hazard, Willian (Pedro 63')