The lively striker finished in style from Mark Yeates' fine pass in the 10th minute of a competitive clash at Selhurst Park.
The lads will now face a trip either Luton Town or Middlesbrough in the last eight.
It was a fine team performance but the plaudits must go to the defensive unit, not least the central three of skipper Danny Foster, Ricky Dobson and David Tyrie, who snuffed out the not unconsiderable threat of Wayne Routledge and Teeratep Winothai as the home side were restricted to a handful of clear-cut chances.
Indeed, Paul Rutherford barely had a save to make as white shirts threw themselves in front of every shot on goal.
After some early sparring and a couple of close offside calls against Jamie Slabber, Palace had the first serious attempt on goal as Ben Watson fizzed a 20-yarder just wide after seven minutes.
Slabber then out-muscled James Gibson in the left channel of the area but, under severe pressure, slid the ball wide from a tight angle.
The opener wasn't far away though. Yeates threaded through the perfect pass for Malcolm to outpace two defenders and tuck the ball past the advancing Lance Cronin. Ten minutes on the clock and the perfect start.
Palace responded well and Tariq Nabil's low drive from the edge of the box struck the foot of the post before being scrambled away. The same player had a better chance from the ensuing corner but blazed over.
Both sides tried to play football on a lively surface but chances were few and far between for the rest of the half as our defence coped with the pace and threat of Winothai and Routledge.
At the other end we always looked likely to break through Palace's high defensive line but Slabber, Malcolm and the ever-willing Yeates, breaking from deep, were constantly thwarted by the offside flag.
Palace upped the ante in terms of competitiveness in the second half but, again, there was little in terms of chances in the opening exchanges.
Indeed, it was to our defence's credit that Winothai and Routledge, seen as the danger men beforehand, were substituted on the hour. One of those replacements, Gary Prigent, had the first decent chance of the second half within seconds of coming on but fired high and wide.
Malcolm finally beat the offside flag to latch onto Lee Barnett's pass after 67 minutes but Cronin read his intentions and smoothered the one-on-one chance.
Slabber soon headed inches wide from substitute Jon Black's corner and Gibson's close-range effort was blocked by a combination of Rutherford and Dobson as the game started to open up.
Still Palace failed to break down our defence and it wasn't until the 85th minute that the next chance arrived. From a short corner routine, Watson completely missed his kick, Dobson then blocked from Nabil and finally Watson's overhead bounced tamely wide. That was that though as Pat Holland's lads marched into the last eight.
Line-ups: Palace: Cronin, Wilson, Gibson, Borrowdale, Simpson, Routledge (Berry, 61), Heeroo, Nabil, Watson, Soares, Winothai (Prigent, 61). Subs: El-Sahali, Wilkinson, Conroy.
Spurs: Rutherford, Ifil, McKie, Tyrie, Foster, Dobson, Barnett, Malcolm (Price, 78), Slabber, Yeates, Wettner (Black, 65). Subs: Eyre, McKenna, Watson.
Michael Malcolm's early goal booked our Academy's place in the quarter-finals of the FA Youth Cup at Crystal Palace on Thursday night.