Old English sliefe (West Saxon), slefe (Mercian) "arm-covering part of a garment," probably literally "that into which the arm slips," from Proto-Germanic *slaubjon (cf. Middle Low German sloven "to dress carelessly," Old High German sloufen "to put on or off"). Related to Old English slefan, sliefan "to slip on (clothes)" and slupan "to slip, glide," from PIE root *sleubh- "to slide, slip."
Cf. slipper, Old English slefescoh "slipper," slip (n.) "woman's garment," and expression to slip into "to dress in"). Mechanical sense is attested from 1864. To have something up one's sleeve is recorded from c.1500 (large sleeves formerly doubled as pockets). Meaning "the English Channel" translates French La Manche.
Rejoice or exult secretly, hide one's amusement, as in When she tripped over her bridal train, her sister couldn't help laughing up her sleeve. This expression replaced the earlier laugh in one's sleeve, used by Richard Sheridan in The Rivals (1775): “'Tis false, sir, I know you are laughing in your sleeve.” The expression, which alludes to hiding one's laughter in big loose sleeves, was already a proverb in the mid-1500s.
see card up one's sleeve; laugh up one's sleeve; roll up one's sleeves; wear one's heart on one's sleeve.