Tyrique George made the most of Enzo Maresca’s half-time reshuffle to set on their way towards a place in the semi-finals.
Youngster George, who joined the club as an under-8, struck his first senior goal four minutes into a much-improved second period having moved into the No.10 role vacated by who was substituted after a dour opening period.
And Noni Madueke, Palmer’s replacement, swiftly followed up with an impressive brace to deflate a spirited but limited Legia challenge, though his second came 25 seconds after Christopher Nkunku fluffed his lines from the penalty spot.
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Legia did offer an early sign of adventure when Rafał Augustyniak’s free header was blocked and Luquinhas fired the rebound over.
But the first-half pattern was soon set: slow, sideways passing from the visitors against unfancied hosts defending with a bank of four in front of a five-man defence.
Chelsea’s only sign of threat in the opening half hour came from set pieces with Tosin Adarabioyo and Benoit Badiashile a mild nuisance in the air.
But it was not until Kiernan Dewsbury-Hall sent a curling effort on target after 34 minutes that Legia stopper Kacper Tobiasz was called into duty.
His save was a solid one but rather than use that sight of goal as a moment to lift the intensity, Chelsea returned to patient keeping of possession.
Half-time arrived with them having 72 per cent of the ball, though Maresca had seen enough.
In addition to removing star man Palmer, still easing his way back following injury, centre-back Tosin was hauled off for Levi Colwill.
And Chelsea instantly began to play with more purpose.
George’s opener was the result of alertness and composure. He reacted first when ’ stinging low strike from 20 yards was parried and the finish was well-taken.

From that point on it was one-way traffic and Madueke gratefully received a pass from Jadon Sancho, took a touch and sent a left-footed finish home.
Madueke added further insurance when tapping in a fine assist from Jadon Sancho, who weaved in from the left before putting the ball on a platter.
That arrived right after Nkunku had a spot-kick saved following a brutal hack from Patryk Kun on Dewsbury-Hall.
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