The Seasiders, targeting a first league victory since a 1-0 triumph at Ilkeston Town in early September, were unable to muster a goal despite creating numerous opportunities at the Towbar Express Stadium, with James Harrison going close for the hosts in the second-half with what was Whitby's best chance of the evening.
Gary Liddle's men, though, were resolute in defence when required and restricted the visitors, fresh from a 1-0 win over FC United of Manchester, to very few chances as they recorded just a fourth clean sheet of the season.
Six changes were made from Saturday's defeat against Stocksbridge Park Steels at Bracken Moor, with Frankie Whelan handed a start prematurely after suffering a hamstring strain in September's 1-1 draw with Stockton Town, while Sam Collins was awarded his first start since August's 5-2 loss at Hyde United after a positive second-half display from the bench during Saturday's reverse.
The Prescot defence were forced into early action as 21-year-old winger James Harrison's fierce strike from the edge of the penalty area was well blocked.
Harrison, who is yet to score a competitive goal since his summer arrival from West Auckland Town, went close to opening his goalscoring account for the Seasiders after his flicked effort from a tight angle, following neat link-up play involving himself and Layton Watts, was blocked by Prescot goalkeeper Calvin Hare.
Whitby were fortunate to remain on level terms as James Foley mis-hit a low effort into the hands of Shane Bland when he perhaps should have caused greater concern for the Whitby goalkeeper.
Whelan was unable to head the hosts into the lead before the half-hour mark as the former Middlesbrough centre-back glanced over Watts' free-kick from midway inside the visitor's half, directing his looping attempt over the crossbar.
In the second-half, Whitby continued their search for an opener, but Donald Chimalilo curled narrowly wide from distance with Hare looking stranded and beaten.
Chimalilo was unable to make a goalscoring impact as the Zambian attacker, who has four Whitby goals to his name so far this season, completed a mazy run to the edge of the box before finding the hands of Hare with a tame effort.
There was an opportunity for Michael Woods to find the net with a volleyed attempt, however, when leaning back, he struck off target.
With the full-time whistle nearing, Prescot were let off as Harrison fired wide on the rebound after Chimalilo had initially been denied by Hare in the visitor's goal.