A closer translation would be "let"; for instance "lasciami stare" (let me be) or "lascialo andare" (let him go). In this case it would be "you cannot let me become sad", but the intended meaning might indeed be not allowing me to become sad. The online dictionary wordreference.com does list allow together with let.
Let's put it like this: the Italian definition of "lasciare" (http://www.treccani.it/vocabolario/lasciare/) has a pretty long number of meanings; among those 6.a: "Seguito da un infinito (o da che e il cong.), è per lo più sinon. di permettere" (followed by infinitive, or "che" and subjunctive, it's mostly synonymous with allow). 6.b has a number of locutions where "let" is more appropriate, as in "l. andare qualcuno, non trattenerlo, concedergli di partire" (letting someone go, not stopping him, allowing him to leave).
Well, being a beginner, I probably have it all wrong, but the given translation is (Google translate) 'Non puoi lasciarmi diventare triste'. That being said, being bilingual French and English, I see this as 'You cannot leave that I become sad'. Either way, I don't seem to get close
Sorry, just wondering, but how are you bilingual French and not know the meaning of 'laisser'? Because it has exactly the same meanings as 'lasciare': e.g. "laisse-moi expliquer" (FR) = "lasciami spiegare" (IT) = "let me explain" (EN). English even uses "laissez faire" (let [them] do [it]) as the name of an economic policy.