Nelson and fellow sophomore Tyler Honeycutt, who finished with 14 points, did most of their damage in the second half, after the Bruins found themselves training by eight points at halftime in the Pac-10 opener for both schools.
Nelson was held to eight points in the first half and Honeycutt, back from a one-game absence because of a sprained shoulder, scored 12 of his points after the break.
Honeycutt, UCLA's leading scorer, ignited a 20-6 run that brought the Bruins (9-4) back from a 37-29 halftime deficit. Honeycutt scored eight of the Bruins' 20 points during that run, with six coming on two 3-point baskets.
Washington State (10-3) lost its second in a row.