Independiente, who will play Huracan for a spot in the final, struggled in the first half but came alive in the second with Angulo sealing the victory with a superb finish just after the hour-mark at La Bombonera.
As Boca looked to make a quick break from an Independiente corner, the Colombian intercepted the ball and raced back into the box before slipping inside defender Ayrton Costa and firing high over the goalkeeper's head.
Angulo had said before the game that he would score against Boca and was happy to deliver on that promise.
"A promise is meant to be kept, right?" he told TNT Sports.
"A beautiful goal (I heard) from my teammates, I didn't get to see it clearly in my own excitement."
Boca had a goal ruled out for offside in the 15th minute and squandered a golden opportunity towards the end of the first half when Miguel Merentiel's whipped cross found Carlos Palacios all alone in front of goal but he somehow missed the tap-in.
The introduction of Ander Herrera and Edinson Cavani in the second half did little to turn the tide for Boca as frustrated fans made their discontent clear.
"There's a lot of pain... we had two top players from the bench but we couldn't pull it off," interim Boca coach Mariano Herron told reporters.
Herron had replaced the sacked Fernando Gago last month and doubt surrounds his future after Boca failed to reach the final.
"About the matches I coached, I take responsibility," he added. "I don't know if it's a failure, but I didn't achieve my objectives, which was to reach the June 1 final."
Boca next face Atletico Tucuman in the Copa Argentina before starting their Club World Cup campaign in the United States.
San Lorenzo also secured a semi-final spot after beating Argentinos Juniors 8-7 on penalties following a 1-1 draw on Monday. They will face the winner of the quarter-final between River Plate and Platense on Tuesday.