David Wear and Lazeric Jones added 15 points each for the Bruins (13-10, 6-5 Pac-12).
Smith was the catalyst for the Bruins, scoring six points in 2 minutes midway through the second half and holding the Cougars' top scorer, Brock Motum, scoreless for much of the half.
Motum, who led all scorers with 25 points, scored 19 in the first half.
"Motum is a great player," UCLA coach Ben Howland said. "And we knew that. He had 34 two games ago, 26 against USC. He's one of the better players in the conference."
The Bruins (13-10, 6-5) were down 34-29 at the half, due in part to Motum. But they dropped into a zone, and that was the difference.
"Our zone defense really helped us today," Howland said. "(Washington State was) trying to screen high. Jerime (Anderson) did a good job fighting over the top of those screens. ... Both their bigs, Motum and Lodwick, are such good perimeter shooters it's hard to match up."
With 11:40 left, the Cougars led 49-45. But they scored only one basket in nearly 9 minutes. During that span, the Bruins outscored Washington State 13-2 and took a 58-51 lead with 4:49 left.
Lodwick broke the run when he hit a 3-pointer, narrowing the gap to four. Howland said when Lodwick hit that shot, he told his players they'd have to hit their foul shots down the stretch.
The Bruins didn't hit them all, but they hit enough to seal the win, going 5 of 8 in the final 30 seconds.
The Cougars outrebounded the Bruins 37-30, but UCLA took advantage of sloppy play by the Cougars, outscoring them 18-3 off turnovers.
"I thought the guys played hard, but sloppy turnovers really cost us," Washington State coach Ken Bone said. "We don't have a defense for those turnovers. . They end up in breakaway layups, and that really hurt us."
The win gave the Bruins a split in Washington for the week, and Howland said while he wanted to leave with two wins, getting at least one was vital.
"I would have liked to be 2-0, and I felt we had a good opportunity to do that, but it's important for us to get this win on the road," he said. "And we needed it desperately. We're going on a real tough stretch here."