toothed cutting tool, Old English sagu, from Proto-Germanic *sago "a cutting tool" (source also of Old English seax "knife," Old Norse sög, Norwegian sag, Danish sav, Swedish såg, Middle Dutch saghe, Dutch zaag, Old High German saga, German Säge "saw"), from PIE root *sek- "to cut" (source also of Latin secare "to cut").
Old English toð (plural teð), from Proto-Germanic *tanthu- (source also of Old Saxon, Danish, Swedish, Dutch tand, Old Norse tönn, Old Frisian toth, Old High German zand, German Zahn, Gothic tunþus), from PIE root *dent- "tooth." Plural teeth is an instance of i-mutation.
The loss of -n- before spirants is regular in Old English, Old Frisian, and Old Saxon: compare goose (n.), five, mouth (n.). Also thought, from stem of think; couth from the stem of can (v.1); us from *uns.
Application to tooth-like parts of other objects (saws, combs, etc.) first recorded 1520s. Tooth and nail as weapons is from 1530s. The tooth-fairy is attested from 1964.