Miley climbed highest to connect with substitute Sandro Tonali's corner at the Gallowgate End to finally break Fulham's resolve in this tense last eight clash.
Summer signing Yoane Wissa made his full debut for the Magpies against the Cottagers at St. James' Park and opened the scoring in the 11th minute.
Fabian Schär's crossfield ball to Jacob Murphy allowed the winger to drill the ball into the box from the right, and although goalkeeper Benjamin Lecomte pushed it out, Wissa gambled and fired home.
But Fulham quickly levelled when Saša Lukić stole in unmarked to head Antonee Robinson's delivery beyond Aaron Ramsdale.
Kevin then squandered a decent opportunity for the visitors, shooting straight at Ramsdale, before Bruno Guimarães was off target for the home side, whose starting XI featured six changes to the side beaten at Sunderland in the Premier League at the weekend.
In the 37th minute, a deflected Guimarães drive was palmed away by Lecomte as the sides went in level at the break despite some late United pressure, with a goalbound Murphy effort blocked by Joachim Anderson.
The second half took some time to get going but United did have the ball in the net in the 65th minute when Guimarães played a lovely reverse pass for Murphy, whose low centre went in off Fulham skipper Anderson - but Scott Ledger's flag was already on its way up.
Fulham's Emile Smith Rowe then made space for a shot but could only fire straight at Ramsdale from the edge of the box.
Tonali and Nick Woltemade were introduced in the 72nd minute as Eddie Howe's men pushed for a winner. Tino Livramento also limped off to be replaced by Alex Murphy before Anthony Gordon and Anthony Elanga joined the action but a penalty shoot-out loomed large.
In the first of six minutes of additional time, Miley was denied by a stunning Lecomte save - but from the resulting corner, the teenage midfielder won the tie and kept United's defence of the Carabao Cup alive.
Newcastle United: Aaron Ramsdale, Fabian Schär, Yoane Wissa (Nick Woltemade 72), Harvey Barnes (Anthony Gordon 85), Malick Thiaw, Tino Livramento (Alex Murphy 76), Jacob Murphy (Anthony Elanga 85), Joe Willock (Sandro Tonali 72), Bruno Guimarães (c), Jacob Ramsey, Lewis Miley
Subs not used: John Ruddy, Joelinton, Leo Shahar, Sean Neave
