early 15c., dismissen, "release from court restraint or legal charges;" late 15c., "remove from office, service, or employment," apparently from Latin dimissus, past participle of dimittere "send away, send different ways; break up, discharge; renounce, abandon," from dis- "apart, away" (see dis-) + mittere "send, let go" (see mission). Meaning "send away, order or give permission to depart" is from 1540s.
The prefix was altered by analogy with many dis- verbs. Middle English also had dismit, in the same sense (late 14c.), with altered prefix but directly from the Latin verb, and dimit. Related: Dismissed; dismissing.