The Crew dominated possession early on and LAFC's defence succumbed when a Diego Palacios handball in the 31st minute led to a goal by MVP Hernandez, who slotted home his penalty.
The Crew doubled their lead four minutes later when Malte Amundsen made one of the best assists of the season with a long pass to Yeboah, who put the ball away to delight the rain-soaked, gold-and-black clad fans at Lower.com field.
LAFC, whose style relies on counter-attacking, never got into gear in the first half and Crew keeper Patrick Schulte was not forced to make a save until close to half-time.
After the break, Carlos Vela, playing what could be his final game for LAFC, tried to capitalise on a defensive miscue but his shot went directly into the arms of Schulte.
LAFC finally responded in the 74th minute when Jesus Murillo stole the ball in the Crew's half and found an open Denis Bouanga, whose first attempt hit Schulte in the face.
The golden boot winner fired the rebound home to cut the deficit but the Crew defence held out to ensure victory on a rainy day in Central Ohio.
"Words can't describe it," said midfielder Aidan Morris.
"We've made so many sacrifices. What we do in this profession is a lot of work. It's a lot of days away from your family, it's a lot of travelling, so to have it end like this is a dream come true.
"That's what we do at this club and we'll go for it again next year."
The win is the latest in a turnaround for the Crew, who were one of the league's original franchises.
Previous owner Anthony Precourt in 2017 threatened to move the franchise to Austin but a grassroots movement helped keep the team in central Ohio.
Cleveland Browns owner Jimmy Haslam became the club's owner in 2019 and the team began playing at their new stadium two years ago.
The Crew won their first title in 2008 and second in 2020 in front of a greatly reduced crowd due to the pandemic. Only three players from that squad were in the lineup on Saturday.
Under head coach Wilfried Nancy, in his first season in charge, the Crew's risk-taking attack scored more goals than any other team during the regular season (67) and post-season (15).
The Crew beat Atlanta United, Orlando City SC and top seeds FC Cincinnati to reach Saturday's final.
The match capped a banner year for MLS, which launched a new team in St. Louis, set attendance records, began a 10-year media rights deal with Apple and welcomed all-time great Lionel Messi to Inter Miami.
The US will be a focal point for football in the coming years as it is set to host the Copa America in 2024, the men's World Cup with Mexico and Canada in 2026, possibly the women's World Cup in 2027 and the 2028 Olympics in Los Angeles.
