The 26-year-old made the breakthrough in the 47th minute before heading a second on the hour, and although Croatia halved the deficit in the 75th minute through Mario Pasalic, the visitors could not find a second late on during a disappointing performance.
The result has moved Wales into second spot in Group D, above Croatia on the head-to-head record, while Rob Page's side are six points behind leaders Turkey with a game in hand.
Wales' final two games in the section are against Armenia and Turkey in November, and the team have now given themselves a real chance of claiming a top-two position in the section.
Wales had a brilliant chance to make the breakthrough in the 12th minute of the contest when Neco Williams found Wilson in the penalty area, but Croatia defender Domagoj Vida was on hand to make an excellent block.
Wilson came close again in the 23rd minute with a brilliant free kick, but Croatia goalkeeper Dominik Livakovic stretched across his goal to turn the attacker's effort wide of the post.
Wales were the team on the front foot in the first period, with Williams bringing another smart save from Livakovic in the 30th minute, and although Croatia had more of the ball as the first period drew to a close, there was very little of note from the visitors in terms of an attacking threat.
The hosts made their pressure count in the 47th minute, with David Brooks picking out the run of Wilson, who expertly lifted the ball over the onrushing goalkeeper to spark jubilant scenes inside the stadium.
Wilson was at it again on the hour, with the Fulham attacker finding the back of the net with a wonderful header from a Daniel James cross, with the former only entering the field three minutes earlier in place of Brooks.
James was lively and had a chance to make it 3-0 in the 68th minute after being released inside the box, but his curling effort was always missing the target.
Croatia managed to halve the deficit in the 75th minute through Pasalic, who headed deflected a corner from Luka Modric into the back of the net from close range.
Modric was the one driving Croatia forward in search of a leveller, with the experienced midfielder blazing over the crossbar in the 82nd minute, but Wales managed to navigate their way through a testing final period to pick up a huge three points in the section.