Old English swat "perspiration, moisture exuded from the skin," also "labor, that which causes sweat," from Proto-Germanic *swaitaz "sweat" (source also of Old Saxon, Old Frisian swet, Old Norse sveiti, Danish sved "sweat," Swedish svett, Middle Dutch sweet, Dutch zweet, Old High German sweiz, German Schweiß), from PIE *sweid- (2) "to sweat" (source also of Sanskrit svedah "sweat," Avestan xvaeda- "sweat," Greek hidros "sweat, perspiration," Latin sudor, Lettish swiedri, Welsh chwys "sweat").
A widespread set of Slavic words (Polish, Russian pot "sweat") is from Old Church Slavonic potu, related to peku "heat," cognate with Latin coquere.
The Old English noun became Middle English swote, but later altered to the current form under the influence of the verb. Sweat of (one's) brow as a symbol of toil is from Genesis iii.19. Sweat equity is from 1968. Colloquial no sweat "no problem" attested from 1963.