Old English blæst "a blowing, a breeze, puff of wind," from Proto-Germanic *bles- (source also of Old Norse blastr, Old High German blast "a blowing, blast"), from PIE root *bhle- "to blow."
Meaning "explosion" is from 1630s; that of "noisy party, good time" is from 1953, American English slang. Sense of "strong current of air forced into a furnace to accelerate combustion for iron-smelting" (1690s) led to blast furnace (1706) and transferred American English sense in full blast "the extreme" (1836). Blast was the usual word for "a smoke of tobacco" c. 1600.
Old English blæstan "to blow, belch forth," from Proto-Germanic *bles- (source also of German blasen, Gothic blesan "to blow"), from PIE root *bhle- "to blow." From 16c.-19c., often "to breathe on balefully, cause to wither, blight, prevent from blossoming or maturing." Meaning "to blow up by explosion" is from 1758. Related: Blasted; blasting.