Our London rivals - the cup holders and finalists in six of the last seven seasons - controlled the first period and took a 2-0 lead in at the break courtesy of headers from Reece James and Ike Ugbo.
We produced some good play in flashes but the visitors were in command and Austin made a string of top-drawer saves to deny all comers in the opening stages of the second half, including one with his face to thwart Ugbo from point-blank range.
But as the game opened up in the last 20 minutes, we started to threaten more in the final third and found an 88th-minute lifeline when substitute Jack Roles played in Sterling to finish clinically into the bottom corner.
The drama wasn't over, though. Tashan Oakley-Boothe was penalised for his challenge on Harvey St Clair in the box and Austin found his resolve being put to the test one final time as he faced McCormick's spot kick.
And the 18-year-old was relieved to see the penalty strike the base of his right-hand post, meaning we're 2-1 down and within touching distance of the Blues in advance of Saturday's second leg at Stamford Bridge.
GOAL: SPURS 0-1 CHELSEA - REECE JAMES - 34mins
Chelsea set themselves up to cut out our attacks in midfield and went close inside seven minutes when Mason Mount's free-kick struck the outside of the post. Ugbo and Dujon Sterling had near misses for the visitors before they made a 34th-minute breakthrough, although centre-backs Japhet Tanganga and Jon Dinzeyi will have been devastated with it after their initial solid defending. Tanganga made a vital block from Sterling in the box, Dinzeyi got in the way of Jacob Maddox's follow-up and Tanganga was there once more to put Callum Hudson-Odoi's effort behind. From the resulting corner, though, James made a near-post run and his up-and-under header sailed beneath the crossbar to give Chelsea the lead.
GOAL: SPURS 0-2 CHELSEA - IKE UGBO - 40mins
The Blues doubled their advantage six minutes after their opener when Sterling volleyed a high ball back across goal from the right and Ugbo was on hand to steer in a header from eight yards out.
SAVES: AUSTIN HEROICS KEEP US IN IT - 50-58mins
We made a brighter start to the second period but were indebted to Austin for several crucial stops. He got down low to his left to deny Mount from the edge of the box on 50 minutes, then dived left once more to beat away Hudson-Odoi's effort from slightly further out five minutes later. Within 60 seconds of that, Ugbo's close-range effort hit the keeper in the face and he was there again shortly before the hour mark to claw away James' sweetly-struck volley.
Above: Tashan Oakley-Boothe in action.
CHANCE: TANGANGA GETS FORWARD BUT FLUFFS HIS LINES - 69mins
We were starting to threaten more as the game progressed and on 69 minutes, Tanganga carried the ball forward with purpose and exchanged passes with Sterling, only to scuff his effort at goal with the ball cleared from inside the six-yard box. Sterling's subsequent effort deflected through to Jamie Cumming in the Blues' net while, a few minutes later at the other end, Austin was at his best again to deny Ugbo one-on-one with a low save, before getting his fingertips to George McEachran's shot on the second phase of play – brilliant goalkeeping from the youngster.
GOAL: SPURS 1-2 CHELSEA - KAZAIAH STERLING - 88mins
The game was much more open in the latter stages and after Dylan Duncan's looping header dropped just off target, we got a goal back on 88 minutes when Roles picked out Sterling in the box and he finished coolly across the keeper and into the bottom corner.
PENALTY MISSED: MCCORMICK UNABLE TO CONVERT FROM THE SPOT - 90+4mins
We were chasing a leveller in stoppage time but in the second added minute, the referee pointed to the spot after Oakley-Boothe's challenge on St Clair down the side of our area. After a slight delay, McCormick took the penalty and, while Austin dived the right way, it crashed off his right-hand post and away to safety, much to the relief of the 2,533-strong crowd at the Lane that included academy graduates Harry Winks, Josh Onomah, Cameron Carter-Vickers and Kyle Walker-Peters.
JOHN MCDERMOTT, HEAD OF PLAYER DEVELOPMENT, SAID
"It's mixed emotions tonight. It's a stage where you want to play and as the home team you want to be more expansive than we were tonight. We were wasteful in possession too many times and credit to Chelsea, I thought they were strong and they showed their quality. We've got to take credit that we stayed in the game, Brandon made some big saves and in the last 20 minutes it was a more even game but we've got to do better in possession and impose what we're about rather than be reactive to the opponents. I don't think we showed our best – our players are better than that performance, certainly in possession, but the fact that we're still in the tie is credit to the strong mentality that this group has. I feel pretty confident that we will play better than that at Stamford Bridge.
"Brandon exhibited there what he exhibits in training. He's an excellent shot-stopper. He works with Pau Lopez, with Michel Vorm and Hugo Lloris and he has that speed of reflex. What's disappointing is that more players didn't exhibit what they've shown in training and in previous games but with the fact that we've got another game on Saturday now, it's a real opportunity to put right some of the things that weren't good enough tonight."
Spurs: Austin, Eyoma, Tsaroulla, Marsh (c, Roles 74), Dinzeyi, Tanganga, Shashoua (Duncan 46), Skipp (Griffiths 68), Sterling, Oakley-Boothe, Bennetts. Substitutes (not used): Hinds, Whiteman (GK).
Chelsea: Cumming, James, T Chalobah, Grant, Castillo, Uwakwe (McCormick 72), Sterling, Maddox, Ugbo (St Clair 83), Mount (c), Hudson-Odoi (McEachran 62). Substitutes (not used): Guehi, Bulka (GK).
Goals: Spurs - Sterling 88; Chelsea - James 34, Ugbo 40.
Yellow cards: Spurs - Marsh 46, Tanganga 90+3; Chelsea - Mount 90+3.
Referee: Lee Swabey.
The second leg of this semi-final tie comes up quickly, as we travel to Stamford Bridge on Saturday (March 18) for a 5.15pm kick-off.