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Playing his 19th Test, Campbell completed his third fifty and first in the Caribbean, ending on 58 off 67 balls (9x4s, 1×6).
Blackwood, who joined Campbell late on day three with West Indies in tatters at 9-3, played a calm innings of 26 not out off 53 balls (2x4s).
While the West Indies batsmen were tested on the third afternoon, the fourth morning was a cruise as Campbell and Blackwood batted with ease and comfort to knock off the remainder of the target.
Khaled Ahmed ended with 3-27.
Kemar Roach, with figures of 2-21 and 5-53, was named Man-of-the-Match.
With his performance, Roach drew level with Michael Holding on 249 Test wickets and also continued his fine form at the Sir Vivian Richards Stadium in Antigua, reaching 50 Test wickets at the venue.
The second and final Test will take place in St. Lucia from June 24.
