Spurred on by their home support, Las Palmas made a quick start to take a surprise fifth-minute lead against their heavyweight opponents.
Oli McBurnie showed great centre-forward play to feed Alberto Moleiro, who neatly created the space to drill a low strike into the far corner.
At the opposite end of the pitch, former Barcelona goalkeeper Jasper Cillessen made two impressive stops from Federico Valverde and Antonio Rudiger to send his team into the half time break with a slender advantage.
Real Madrid started to pile the pressure on Las Palmas after the half-time interval, and their persistence ultimately forced the hosts into a mistake.
Alex Suarez was penalised for handling the ball in the box, giving Vinicius Junior the opportunity to thump home a 69th-minute spot-kick.
While Real Madrid continued to push forward in search of a winner, it was the hosts who went agonisingly close to snatching all three points, with Viti seeing his late strike chalked off for offside.
Los Blancos ultimately had to settle for a point, leaving them in fourth place and four points adrift of leaders Barcelona.
Real Madrid's star summer signing Mbappe will also be frustrated, with the forward still waiting to score his first league goal for his new club.
As for the hosts, they will be pleased to take their first point off Real Madrid since playing out a 3-3 in March 2017.
LAS PALMAS VS. REAL MADRID HIGHLIGHTS
Moleiro goal vs. Real Madrid (5th min, Las Palmas 1-0 Real Madrid)
The hosts take a surprise lead! McBurnie collects the ball on the left flank, holds it up and waits for the right moment to feed Moleiro. The 20-year-old evades Aurelien Tchouameni and Eder Militao before sending a low shot into the far corner.
Vinicius goal vs. Las Palmas (68th min, Las Palmas 1-1 Real Madrid)
Penalty to Real Madrid. Arda Guler prods the ball onto Suarez's left arm and the referee quickly points to the spot. Vinicius steps up to thump the penalty past the outstretched Cillesen.
Viti disallowed goal vs. Real Madrid (89th min, Las Palmas 1-1 Real Madrid)
Viti thinks he has won it late on. However, the substitute's emphatic strike is ruled out for an offside infringement from the initial knockdown.
MAN OF THE MATCH - ALBERTO MOLEIRO
Moleiro is highly rated at Las Palmas and has compared to Barcelona midfielder Pedri, who came through the youth ranks at Los Amarillos.
The youngster showcased his talent against the European champions, utilising his close ball control to create the space to fire a left-foot strike past Thibaut Courtois.
In addition to his goal, Moleiro achieved a 96% pass competition rate and won eight of his duels - more than any other player on the pitch.
LAS PALMAS VS. REAL MADRID MATCH STATS
Possession: Las Palmas 44%-56% Real Madrid
Shots: Las Palmas 7-25 Real Madrid
Shots on target: Las Palmas 2-8 Real Madrid
Corners: Las Palmas 2-6 Real Madrid
Fouls: Las Palmas 16-13 Real Madrid
BEST STATS
WHAT NEXT?
After being held to a frustrating draw, Real Madrid will attempt to return to winning ways in Sunday's home clash against Real Betis.
Meanwhile, Las Palmas will be buoyed from their hard-fought point, and they will now look to claim their first win of the season in their away meeting with Alaves this weekend.