Words such as fly, gnat, and midge, which all derive from OE, were used to described a variety of two-wing insects, biting and non-biting. Gnat is the equivalent you're looking for, though they were a type of "flie". From the OED entry on "mosquito":

These insects were formerly known in Britain as gnats, though that term is now taken to include many non-biting forms.

Mosquito appears to have entered the language as a word to describe New World biting flies and was indeed borrowed from the Spanish. Many of the early citations below take care to compare the "muskito" to the "gnatt" for the reader's benefit.

1572 H. Hawks in Writings & Corr. Two Richard Hakluyts (1935) i. 97 A certeine gnat or flie which they call a Muskito, which biteth both men and women in their sleepe.

1585 R. Grenville in Hakluyt's Voy. iii. xxiv. 734 The 29. day wee set saile from Saint Iohns, being many of vs stoong before vpon shoare with the Muskitoes.

1589 M. Philips in R. Hakluyt Princ. Navigations iii. 568 Wee were also oftentimes greatly annoyed with a kinde of flie,..the Spanyards called them Musketas.

1600 M. Sutcliffe Briefe Replie to Libel iii. vii. 35 He is like a flye, or rather, because he speaketh so much for Spaniards, a Spanish mosqueta.

1623 R. Whitbourne Disc. New-found-land 99 A very little nimble fly..which is called a Muskeito.

1634 W. Wood New Englands Prospect i. xi. 46 The fourth is a Musketor which is not unlike to our gnats in England.

1655 E. Terry Voy. E.-India 123 In the night we were..very much disquieted with another sort [of fly] called Musqueetoes.

1665 T. Herbert Some Years Trav. (new ed.) 128 Howbeit the Muskitto's or Gnatts pestered us extremely.

This may be a better question for /r/linguistics