The Chilean league title holders won the two-legged final 4-0 having upset LDU 1-0 with a goal by Vargas, the competition's top scorer with 11, in Quito last week.
Vargas opened the scoring in the third minute to put "La U" two-up on aggregate and set the Ecuadorian team, tournament winners in 2009, a huge task to get back into the tie.
He added a brilliant third goal three minutes from time after dribbling past two defenders and chipping over goalkeeper Alexander Dominguez to ecstatic cheering from a capacity 45,000 crowd at the Estadio Nacional.
LDU defender Jorge Guagua was shown a straight red card in the 68th for elbowing substitute Gustavo Lorenzetti, who 11 minutes later scored the home side's second goal.
Universidad de Chile had Argentine midfielder Matias Rodriguez sent off for a second booking six minutes from the end.
"I never imagined I'd achieve this, to be in a squad that wins such an important cup is to touch the sky with your hands," Universidad de Chile's Argentine-born striker Gustavo Canales told Fox Sports.
The Chilean team, coached by Jorge Sampaoli who has spent all his coaching career outside his native Argentina, were unbeaten through the tournament, winning 10 of their 12 matches and scoring 21 goals and conceding two.
"We were aware we played a different game (from the other teams), never renouncing the priority to attack," Canales added of Sampaoli's penchant for constant possession and attack.
"La U", unbeaten in 35 matches, won the domestic Apertura title in the first half of the year and have reached the semi-finals of the Clausura.
The Copa Sudamericana is South America's second most important club competition after the Copa Libertadores, the region's champions league which Universidad de Chile's arch-rivals Colo Colo won in 1991.
