Hawkins shrugged off the challenge of centre back Dave Robinson before beating Ryan Price at the near post with a low drive from 12 yards out.
Tamworth had thought they had salvaged a point after Bob Taylor had scored three minutes earlier after challenging Grecians' keeper Paul Jones to a high cross and stabbing home the loose ball.
A draw would have been just reward for a spirited second half comeback by the Lambs.
They were chasing the game after Kezie Ibe had put the visitors in front after 23 minutes when he reacted the quickest to snap up a rebound after Price had pulled off a one handed save to deny Les Afful.
Tamworth applied the pressure in the second half and forced a series of corners.
They had two goals ruled out for offside and manager Mark Cooper must have thought the equaliser wouldn't come.
Firstly, Taylor headed home after giant defender Matt Redmile had outjumped the City defence to head a Lee Colkin free kick across goal just after the restart.
The flag of linesman Carl Henry again cut short the Tamworth celebrations in the 72nd minute when Taylor flicked on a high ball and teenager Kyle Storer slipped the ball past Jones.
"The officials have cost us dearly and we feel cheated after a spirited second half performance," said a disgruntled Cooper after the game.