The 30-year-old Street (1-0) has 24 saves in 25 opportunities and a 1.09 ERA in 33 games this season.
He was unavailable to the Padres during their 5-4 home loss to the New York Mets on Friday night.
San Diego (41-55) is in third place in the NL West, 12 games off the pace.
"Losing is a miserable experience," Street said before the game. "I believe in the Padres' ownership. They want to win and they are not content with status quo.
"I blame the players for what's happened here."
In 10 major league seasons with Oakland (2005-08), Colorado (2009-11) and San Diego (2012-14), Street is 35-27 with 258 saves and a 2.87 ERA.
He has converted 109 of his last 117 save chances, the best mark in the majors in that span.
Street, fifth on the active list in saves and 31st overall, said the Padres were upfront with him regarding the trade talks.
"I asked questions and they gave me direct answers," said Street, who was acquired by the Padres after the 2011 season. "They were extremely honest, said they were fielding offers, and asked me to not be offended."
The Angels entered Friday 1½ games behind the league-leading Oakland Athletics in the AL West.
Last month, they acquired Pittsburgh Pirates closer Jason Grilli in a deal that sent Ernesto Frieri to Pittsburgh.
The Los Angeles Angels did major patchwork on their bullpen on Friday, obtaining closer Huston Street, who will bring his dominance to a team that needs it. Story
Frieri had struggled this season for Los Angeles, going just 0-3 with a 6.39 ERA and 11 saves in 34 appearances before the trade. Grilli, meanwhile, has appeared in eight games for the Angels since the deal, going 1-1 with a 1.29 ERA.
Joe Smith has been serving as Los Angeles' closer, collecting 15 saves.
Gott, 21, has split this season between Single-A Lake Elsinore and Double-A San Antonio. In 39 combined relief outings, he is 2-4 with 16 saves and a 3.56 ERA.
He was chosen as a midseason California League All-Star.
Lindsey, a 22-year-old second baseman, was the No. 37 pick by the Angels in the 2010 draft. In four-plus minor league seasons campaigns, he has batted .289 with 43 home runs and 209 RBI.
He was rated as the Angels' top prospect by Baseball America.
Alvarez, 23, has gone 7-4 with five saves in two-plus minor league seasons.
The 20-year-old Rondon was signed as an international free agent in January 2011. The Venezuelan infielder has played in 245 minor league games, batting .300 with 146 runs and 116 RBI.
Morris, 22, has appeared in 28 games -- 18 starts -- in the Angels system, going 7-6 with a 3.43 ERA.
ESPN Baseball Insider Jim Bowden and The Associated Press contributed to this report.