Torje opened the scoring ten minutes into the second period and Marica made it two in the 75th minute as Victor Piturca's men made a perfect start to their Group D campaign.
Estonia were far from outclassed in the first half and took the game to their opponents in the early exchanges, with their record goalscorer Andres Oper firing wide of the target in the eighth minute.
Romania took charge of proceedings as the opening half wore on, but they could not carve out much in the way of clear-cut chances, with Costin Lazar and Gheorghe Grozav among those reduced to hopeful long-range efforts.
Marica was next to try his luck from distance in the 48th minute when he fired in a shot that Estonia goalkeeper Sergei Pareiko pushed wide.
Grozav flashed another effort wide on 52 minutes, but within moments Romania were celebrating the opener, Torje stealing in a the back post to turn in Grozav's cross from the left.
And with 15 minutes remaining the visitors set the seal on their victory, Torje turning provider to pick out Marica and the Schalke striker took aim with a low finish into the left corner.