Dimitri Payet set up both goals for OM at a packed Velodrome, as more than 62,000 fans saw Thauvin grab an opener that had more than a suspicion of handball about it.
Cameroon international Njie then made it 2-0 less than three minutes after coming off the bench in the second half, to leave Marseille firmly in control of the tie ahead of next week's return in Austria.
"It is a good result. We wanted to win and it's done. And doing it without conceding a goal was a good idea," said Marseille coach Rudi Garcia.
"We know we are not through yet. We will go there looking to score at least one goal and win the game. We need to show we deserve to go to the final."
Marseille are desperate to reach a final that will be played in France, at the home of their bitter rivals Lyon, on May 16.
They are aiming to win a second European trophy, a quarter of a century after their triumph in the inaugural Champions League final in 1993.
Garcia's side would perhaps not be the favourites if they did make it through to face either Arsenal or Atletico Madrid, who drew 1-1 in London in Thursday's other game.
However, they have enough quality in their ranks to trouble anyone, even if Salzburg might feel aggrieved at not at least grabbing an away goal.
"We are sad. We think we deserved better. But we have no choice, we are already thinking about the return match," said the Austrian side's German coach Marco Rose, before adding that he would "not cry" about having a penalty appeal turned down in the second half.
Marseille's opener arrived with just 15 minutes played, as Salzburg goalkeeper Alexander Walke came for a Payet free-kick but couldn't get to it, and Thauvin arrived to finish at the back post with a combination of his head and an arm.
Scottish referee Willie Collum did not see anything wrong with the goal, and he also waved away Salzburg's penalty appeals early in the second half when Stefan Lainer went down as he crashed into Maxime Lopez in the box.
Moments after that, Marseille goalkeeper Yohann Pele produced an excellent save to deny Hannes Wolf, and the home side doubled their lead in the 63rd minute.
Substitute Njie released Payet down the left and then controlled the return ball in the box before firing home, sparking jubilant scenes in the vast home crowd.
Norwegian substitute Fredrik Gulbrandsen hit the post with a half-volley at the other end late on, and Salzburg may live to regret not scoring that precious away goal as they look to become the first Austrian side to reach a European final in over 20 years.