But they had to wait until the 64th minute to find the victor today, with Son Heung-Min's superb effort from the edge of the box flying past Palace goalkeeper Julian Speroni to make it 1-0 with just over 25 minutes remaining.
Despite not being at their best, however, Tottenham remained in third position.
Dier was quoted as saying by the BBC: "We knew it would be hard today with players missing, but we wanted to keep the momentum going. It's great to get the three points ahead of the global break and we'll enjoy watching the rest of the football this afternoon", Son said after the game.
Spurs were without Dele Alli, scorer of two goals in the 3-1 win over Real, while goalkeeper Hugo Lloris and his deputy Michel Vorm were both unavailable, handing a debut to 25-year-old Argentinian Gazzaniga, who made crucial saves to earn the Man-of-the-Match vote.
Palace looked the more likely to break the deadlock at the start of the second half and should have taken the lead. Zaha should have punished Spurs when he was clean through but, after going round Gazzaniga, he shot wide with the goal at his mercy.
However, it was the hosts who would score the only goal of the game, with Son producing a superb curling effort to secure a precious win for his side.