Former Marseille and France international striker Gignac swept home the winner six minutes from time at Orlando's Exploria Stadium to complete a dramatic come-from-behind win for the Liga-MX side.
The victory marked the end of years of frustration in the tournament for Tigres, who had been beaten in the final three times since 2016.
"We lost it almost three years in a row and I think the club deserved to finally win it," Gignac said afterwards.
The Mexican side will now advance to the FIFA World Club Cup in Qatar next February following the win.
"We finally achieved an international title," added Gignac, who was named tournament MVP and Golden Boot winner.
"I always wanted to win it even though some say it is not worth much. It is worth going to a Club World Cup, and it is worth playing against the best teams from each continent. Today it is Tigres' turn."
Tigres win extends Mexico's proud record in the competition. Mexican clubs have won the title for the last 15 editions.
But the loss shattered Los Angeles' dreams of becoming the first side from Major League Soccer to be crowned kings of club football in the North America, Central America and Caribbean (CONCACAF) region since Los Angeles Galaxy managed the feat in 2000.
The defeat was even harder to swallow as the MLS side had appeared to be in total control for long periods of the second half after Uruguayan striker Diego Rossi fired them into a 61st-minute lead.
Los Angeles captain Carlos Vela then missed a golden chance to double the MLS club's lead soon afterwards, taking too long to get his shot off and allowing Tigres to block from close range.
At that point, Los Angeles looked by far the likelier winners against a Tigres side who had barely managed a shot on goal throughout.
But the Mexican club were gifted a way back into the match when a defensive mix-up involving goalkeeper Kenneth Vermeer saw Los Angeles needlessly concede a corner with just under 20 minutes left.
From the ensuing setpiece, Hugo Ayala's glancing header beat Vermeer and crept in at the far post.
The goal transformed the contest, with Tigres pouring forward in search of a winner against a weary Los Angeles.
With extra-time looming, Luis Rodriguez went on a jinking run to the edge of the area before squaring for Gignac in space.
The Frenchman made no mistake with the finish, coolly picking his spot to sidefoot into the bottom corner past former Ajax goalkeeper Vermeer.
Tuesday's final marked the climax of a tumultuous tournament which began in February only to be halted abruptly midway through the quarter-finals in March as the Covid-19 pandemic erupted.
The competition resumed last week with all remaining teams playing in a quarantine bubble in Orlando in games played behind closed doors.
Standout matches today