Coventry were a cut above in the first half and oozed confidence once Callum Wilson fired them in front as early as the fourth minute.
Wilson raced onto a Carl Baker pass and coolly rounded goalkeeper Chris Weale before stroking home his sixth goal of the season.
Coventry called the tune for long stages and Weale had to be alert to deal with shots from outside the box from the lively Wilson and Leon Clarke, while he also made a sharp stop low to his right to deny Franck Moussa.
Shrewsbury boss Graham Turner responded by sending on Curtis Main, a striker on loan from Middlesbrough, at the start of the second half and he made an immediate impact.
Main crossed perfectly from the right and Wales under-21 international Bradshaw clinically swept the ball past Joe Murphy from 15 yards.
Shrewsbury then upped the tempo with Aaron Wildig chipping over with just the keeper to beat before Main's late header was tipped over the bar by Murphy as the points were shared.