The 20-year-old defender recovered from giving away two penalties on his Premier League debut last weekend to be the second-half match-winner in Tottenham's seventh away win from eight games this campaign.
Palace remain without a win and an open-play goal at home in the top flight this season, although they could have grabbed a late point from the London derby when Alexander Sorloth shot straight at Hugo Lloris in stoppage time.
There were some nervy moments in the Spurs defence early on, although mostly self-inflicted as the shaky Lloris flapped at a cross which nearly dropped for Jordan Ayew.
Foyth played his goalkeeper into trouble with a slack back-pass that led to a corner from which James Tomkins, a constant weapon from set pieces, could have done better.
Playing against his boyhood club, Andros Townsend worked Lloris with a drilled shot which the Frenchman turned around his post on 15 minutes, as Palace started brightly.
A change of shape midway through the first period helped the visitors take a strangehold of the game, with Dele Alli dropping deeper to influence the game on his first Premier League start in over two months.
Tottenham's 72% possession before the break did not yield a host of chances though, with Moussa Sissoko engineering their best opportunity with a driving run and shot which was thwarted by Mamadou Sakho's block.
Harry Kane lost one of his main supply lines when Kieran Trippier was forced off with a muscle injury, but was gifted a shooting chance by a slip from Luka Milivojevic, with Wayne Hennessey equal to his 25-yard strike.
Spurs were knocking on the door prior to half time, but were always vulnerable from the dead ball and Palace were close to taking the lead on 55 minutes when Tomkins peeled off Alli only to head into the side-netting.
Ironically, a set piece proved to be the source of Tottenham's goal as Kane stooped to beat Martin Kelly, a recent replacement for the injured Tomkins, to the initial header, which hit Milivojevic and looped up invitingly for Foyth to nod home from six yards out.
While Lloris had threatened another costly mistake in a series of errors earlier in the game, the Spurs captain stepped up in the final 10 minutes to make a trio of saves, firstly pushing away a powerful swinger from Jeffrey Schlupp's right foot.
Lloris jumped on Townsend's 84th-minute volley from Schlupp's set up, and then stayed big to block from Sorloth, who should have opened his Premier League account when Milivojevic's deflected shot rolled perfectly into his path.
However, Spurs held on to record a fifth successive 1-0 league win over Palace, who stay out of the relegation zone only on goal difference, while the victors moved level on points with second-place Liverpool and Chelsea.
PALACE (4-4-1-1): Hennessey; Wan-Bissaka, Tomkins (Kelly, 60'), Sakho, Van Aanholt; Kouyate, Milivojevic, McArthur (Sorloth, 70'), Meyer (Schlupp, 67'); Townsend; Ayew
SPURS (4-2-3-1): Lloris; Trippier (Aurier, 24'), Alderweireld, Foyth, Davies; Wanyama, Sissoko; Lamela (Winks, 83'), Dele, Lucas (Son, 71'); Kane