Some 104 lawmakers voted in favor, two were against and seven abstained.
The amendment will be put to a second vote in the parliament in three months' time.
A constitutional amendment must be debated and voted on twice by the Seimas, with an interval of at least three months between the votes, and requires a two-thirds majority, or 94 votes, to be adopted.
Lawmakers from both the ruling bloc and the opposition called on their colleagues to support the amendment granting citizens individual access to the Constitutional Court.
The bill calls for allowing any person who has exhausted all other available means of legal protection to turn to the Constitutional Court over the constitutionality of a piece of legislation if he or she believes that a decision based on the legislation has violated his or her constitutional rights or freedoms.
The Constitutional Court's ruling would serve as a basis for reopening proceedings.
Currently, only the Seimas, a group of lawmakers, the president, the government and courts can directly turn to the Constitutional Court.