Coventry were the better side throughout and made a dream start when ex-Rovers midfielder Franck Moussa volleyed home against his former club in the third minute.
Rovers failed to make the most of their chances in an open attacking game and Iain Hume fired wide when well placed in the 16th minute.
Hume had a lob well saved by advancing keeper Joe Murphy after 13 minutes before Coventry struck again.
Blair Adams produced a dangerous cross from the left and McGoldrick raced in unmarked to score with a glancing header in 26th minute.
McGoldrick missed a great chance to score again when he fired wide with only the keeper to beat before making amends when he slid in to make it 3-0 from close range in the 58th minute.
Rovers continued to battle and pulled one back with a header from Chris Brown in the 76th minute but Adam Barton emphasised City's superiority with a superb curling effort from 20 yards in the last minute.