Here is just one example: in this 2008 paper, the authors use a cleverly optimized computer search to find two examples of sets of $7$ points in the plane (no three collinear, no four on a circle) so that every pair of points is an integer distance apart.
This resolved the question, posed by Erdős, of whether such a set of $7$ points exists.
As Gerry Myerson mentioned in another answer, there are easily thousands of examples in which computation has aided and/or inspired new results.