Debutant goalkeeper Austin, 17, made one save of note in the first half but we were largely untroubled in a game of few chances until the home side turned the screw on the hour mark.
Austin parried away first team regular Jeffrey Schlupp's effort on 27 minutes but that was it from a Foxes side also including Demarai Gray and Polish international Bartosz Kapustka until what turned out to be a match-winning purple patch in the second half.
Darnell Johnson, Gray and Harvey Barnes were all kept at bay by Austin and Kairo Mitchell had a goal ruled out for offside before the brealthrough on 74 minutes with Gray played through and making no mistake one-on-one from the left angle.
Five minutes later and it was all over as defender Elliott Moore rose unchallenged to power home a header from Gray's corner.
CLOSE: GEORGIOU JUST WIDE - 10mins
We started wth plenty of intent in our game and came closest in the opening exchanges as Anthony Georgiou found a pocket of space just outside the penalty area and drove across Daniel Iversen, the goalkeeper relieved to see the shot flash just wide.
SAVE: AUSTIN DENIES SCHLUPP - 27mins
Joe Muscatt found himself under pressure on half-way and looked to go home to Austin, only to leave his backpass short, allowing Schlupp to break clear down the left. Schlupp charged into the box and let fly but Austin was ready and palmed away to his left.
CLOSE: PRITCHARD SLIDING IN - 58mins
We enjoyed a bright spell in the opening 15 minutes of the second half. Shilow Tracey fired over and then a half-chance, Tracey worked a cross from the right and Joe Pritchard was inches away sliding in at the near post. George Marsh then volleyed a yard wide after Iversen punched clear.
Above: Joe Pritchard on the ball at the KP.
SAVE: AUSTIN DENIES JOHNSON - 63mins
Alex Pascanu's cross from the left evaded Schlupp in the box but came out to Johnson who drilled in from 20 yards, Austin got down well to parry. As play continued, Gray flashed a shot over from 16 yards.
GOAL DISALLOWED: LEICESTER - 66mins
The home side thought they'd opened the scoring when Mitchell tapped home after Austin showed a firm right hand to keep out Barnes' shot. The offside flag was up straight away though and the goal ruled out.
SAVE AND OFF THE LINE: LEICESTER FOILED - 70mins
Kapustka's cross-shot from the right angle was turned on target by Gray from point-blank range but kept out by a combination of Austin and Muscatt on the line.
SAVE: AUSTIN DENIES BARNES - 72mins
Leicester were piling on the pressure at this point and Austin was there again to deny Barnes one-on-one from the left angle after he was played in by Kapustka.
GOAL: LEICESTER CITY 1-0 SPURS - DEMARAI GRAY - 74mins
Leicester's pressure finally paid dividends as Gray was played in again left side and this time made no mistake, slotting the ball past Austin into the far corner.
GOAL: LEICESTER CITY 2-0 SPURS - ELLIOTT MOORE - 79mins
Moore ran unchallenged to meet Gray's corner and powered home his header from 10 yards, Austin left helpless this time.
Above: Joe Muscatt sticks close to danger man Demarai Gray.
COACH UGO EHIOGU SAID
"I thought we controlled the game for 50, 60 minutes against an experienced Leicester team with the likes of Schlupp, Gray and Benalouane the big centre-half, without that real cutting edge. We probed and controlled the middle third of the pitch without having the final quality to break them down.
"We lost Anthony Georgiou to an injury just before half-time - we all hope it's not too serious - and when you take everything on board, yes there were positives. Brandon Austin was excellent, clean hands, big moments where he did his job well. He's had college as well this morning so he's had a long day! Joe Muscatt worked his socks off, great application up against someone like Schlupp. He gave us everything and had nothing left in the tank. He'll take heart from that.
"Charlie Owens was reliable, good distribution, Zenon was honest, Cy gave us a little guile but when Schlupp and Gray turned up the gas in the second half you could see the difference. We nullified them for a long time because we got in their faces and kept the ball so it's disappointing overall, the players have put an unbelievable shift in but come away with nothing again."
Leicester City: Iversen, Johnson, Wood (Pascanu 61), Miles (Ndukwu 56), Moore, Benalouane, Kapustka, Barnes, Mitchell (Muskwe 69), Gray, Schlupp. Substitutes (not used): Rowe, Bramley (GK).
Spurs: Austin, Muscatt, Stylianides, Marsh (Tanganga 75), Owens, Maghoma, Tracey, Pritchard (Brown 83), Loft, Goddard, Georgiou (Griffiths 45+2). Substitute (not used): Freeman (GK).
Goals: Leicester City - Gray 74, Moore 79.
Yellow cards: Leicester City - Johnson 24.
Referee: James Adcock.
Ugo will now prepare his squad for their second match in a week in the Premier League 2 as Manchester United visit Hotspur Way on Friday (12noon).