Welsh, who is averaging a double-double this season, made 7 of 14 shots, pulled down 11 rebounds, had four assists and three blocks.
"I got a lot of good looks tonight and I took them," Welsh said.
UCLA (8-3) snapped a two-game losing streak and dealt the Coyotes its first loss in five games. The Bruins improved to 6-1 at home and South Dakota dropped to 11-4 and 4-3 on the road.
Trailing by three at halftime, the Bruins went on a 10-3 run to start the second half and Welsh's basket with 17:25 remaining put UCLA up 46-39. Chris Smith's back-to-back 3-point plays increased UCLA's lead to 69-52 with 7:59 to play and the Bruins led 78-54 on Prince Ali's dunk with 5:01 remaining.
But the Coyotes battled back, going on a 28-6 surge behind Peterson and Armstrong in the final five minutes to cut it to 84-82 with 1.7 seconds left following Armstrong's 3-pointer.
South Dakota got the ball back with 1.5 seconds remaining following a UCLA free throw but Armstrong was unable to get off a 3-point attempt as the game ended. After a lengthy delay the referees reviewed the inbounds play replay and ruled that the Welsh had tipped the ball, allowing the clock to run out and end the game.
"It was the right call," UCLA coach Steve Alford said.
Welsh agreed. "I tipped the ball on the inbounds play," Welsh said. "They made the right call. They made a run on us, but we got the win."
The Bruins defense also stepped up in the second half as they held the Coyotes to seven free throws over a 10:14 stretch midway through.
"The first 36 minutes was beautiful basketball and unselfish, but the last four minutes was selfish," Alford said. "The hardest thing to do is to get up when you've been knocked down. It's a teaching moment for me, not the way I want to do it but they learned a valuable lesson tonight."
The Bruins led 7-2 but then went cold. The Coyotes countered with a 17-5 run to take a 19-12 lead following a basket by Mooney at the 8:46 mark. Hands' 3-pointer just before the buzzer cut the Coyotes' lead to 36-33 at the break.
"I really thought we came out really strong at the start and we were righting the ship," South Dakota coach Craig Smith said. "And to be up by three points was disappointing. I thought we really had control of the game and felt like we should have been up more. We played too much one-on-one and that just fed into their transition and that led to their separation."
South Dakota, the reigning Summit League champions, entered the game with a 48.9 shooting percentage but was held to 41.4 percent (29 of 70).
South Dakota: The Coyotes entered Tuesday's game as one of five NCAA Division I teams to reach 11 wins thus far this season (Duke, Purdue, Albany and Villanova are the others). UCLA has five players averaging in double figures this season: Holiday (17.6), Welsh (12.6), Kris Wilkes (12.3), Hands (12.0) and Ali (10.4).
UCLA SUSPENDED PLAYERS UPDATE
Bruins freshmen Jalen Hill and Cody Riley missed their 11th consecutive game and are still suspended indefinitely as a result of their shoplifting arrest in China in November. LiAngleo Ball, the third player involved in the incident, left UCLA and signed a professional contract to play in Lithuania.
NICE MEETING YOU
Monday was the first time that UCLA and South Dakota have met on the hardwood.
CLOSING IN ON A PLATEAU
UCLA senior center Thomas Welsh and junior guard Aaron Holiday are on the verge of reaching 1,000 career points. Welsh needs 24 points and Holiday 37 points to reach the milestone.
NOT TO SHABBY
South Dakota entered Tuesday's game fifth in Division I in free throws made (224) and fifth in total rebounds (527). The Coyotes are shooting 48.9 percent from the field and have outscored their opponents 544-356 in the paint this season.
South Dakota opens a four-game home stand beginning Thursday night against Northland College.
UCLA travels to New Orleans and will take on Kentucky (9-1) Saturday in the fourth annual CBS Sports Classic. It will be the fifth time the two teams have met in the past four seasons.
Tony Bennett is renowned for his defensive schemes at Virginia. Jay Wright has turned Villanova into an offensive juggernaut. Which coaches rank highest on each side of the ball?
Facing grand larceny charges, Michael Weathers was suspended by Oklahoma State.
The recruiting landscape is heating up. Find out where the top 30 players in the 2019 ESPN 100 are headed next season.