With the likes of Warrington Town and Worcester City having already shocked Exeter City and Coventry City in round one, Simon Grayson's men faced a potential banana skin at West Leigh Park.
However, the League One side showed their class with on-loan Aston Villa striker Robinson particularly impressing with a first-half brace.
Brian Stock's dismissal further hindered the hosts, before Robinson completed his treble late on from the penalty spot after Warren Cummings was shown a straight red card for handballing his initial shot on the line.
Robinson's first came in the seventh minute when he sprung the offside trap to meet Paul Huntington's long ball down the left before squeezing home a well-struck shot at the near post.
Preston continued to utilise the pace and movement of Robinson and the tactic bore fruit again on the half-hour mark when he cleverly side-footed home Paul Gallagher's cross from the right.
Shortly after the break, the hosts saw claims for a penalty waved away by referee Steve Martin when Scott Donnelly was tripped in the area, and Stock's second yellow and subsequent red for a poor challenge on Alan Browne compounded their misery.
Robinson then saw a header cannon off the bar and ensuing scramble led to his shot being blocked by the hand of full-back Cummings who was shown a straight red, the 19-year-old duly scoring the resulting penalty.
Preston can now look forward to a second-round showdown with either Walsall or Shrewsbury Town.
