The Paraguayan outfit will take a commanding lead into the second leg of the final after an impressive home win
The Paraguayan side took the lead in the 23rd minute with a goal of the highest quality, with Silva picking up a short pass from Pittoni near the right wing and setting off on a mazy run towards goal.
He struck a firm left-footed strike from just outside the box that had Atletico goalkeeper Victor well beaten and the ball duly flew in off the post to open the scoring.
Silva was so excited that he took his shirt off to celebrate the goal, drawing a yellow card from referee Nestor Pitana and he will now miss the second leg in Brazil next Wednesday.
Atletico coach Cuca replaced the misfiring Ronaldinho with Guilherme just after the hour mark, but the move did not have the desired effect.
With the match entering the final 10 minutes, Olimpia substitute Juan Ferreyra forced an outstanding save from Victor and Fredy Bareiro somehow missed the follow-up from point-blank range.
Bareiro wasted another good chance with five minutes remaining when he charged into the box but blasted his left-footed effort high and wide.
Atletico defender Richarlyson was then sent off in the 90th minute for his second bookable offence, but the home side had to wait until the fourth minute of stoppage time to double their lead.
Olimpia were awarded a free kick just outside the box and Pittoni stepped up to curl it past Victor with the last kick of the game, although the 'keeper was hard done by, with Atletico substitute Alecsandro getting in his way.
The goal sent the bulk of the crowd at the Estadio Defensores del Chaco into pandemonium, and the Paraguayan club are now well placed in their bid to land their fourth Copa Libertadores title.