The striker had an uncharacteristically poor game, but rose high to nod home an 88th minute Christian Eriksen corner to earn his side a deserved three points, but it was not before the visitors had spurned several good opportunities to take the lead.
Earlier, Kane had a shot saved from the edge of the six-yard box after a defensive mistake in the 10th minute, then sidefooted wide in the 54th with the goal at his mercy after meeting Eriksen's lofted pass.
Wayne Hennessey made a number of fine stops to keep the North Londoners at bay and Kane spurned a couple of decent opportunities.
While Tottenham's performance certainly warranted a goal, it was harsh on Palace, who put in a determined backs-to-the-wall display.
Lucas Moura was soon introduced for his Premier League debut as Pochettino pushed for a victor - Hennessey getting down low to prevent Davies's strike finding the bottom-left corner.
It also marked the England forward's 150th goal in club football, and Dier was full of praise for his record-breaking team-mate. "I thought it wasn't going to be our day because we had a few chances, especially me, and Serge Aurier had one at the end too". Can they get a win against Hodgson's Palace, though?
Hennessey was forced into an excellent save on 73 minutes when Ben Davies' shot looked to be trickling into the bottom corner of the goal, but the keeper was able to push it behind at the last moment.
Kane, as ever, came up with the goods in the end, though, but the 24-year-old claimed he was unaware of the milestone his goal marked.
Roy Hodgson is without 12 players for Crystal Palace as they suffer an injury crisis.
With two Premier League Golden Boots secured in as many years, the England worldwide is chasing down an impressive hat-trick this season.