It is written (John 6:45): "Every one that hath heard of the Father, and hath learned, cometh to Me." Now to learn cannot be without a movement of the free-will, since the learner assents to the teacher. Hence, no one comes to the Father by justifying grace without a movement of the free-will. [ST I-II, Q113, A3]
Of course, this is not to say that we can save ourselves:
The justification of the ungodly is brought about by God moving man to justice. For He it is "that justifieth the ungodly" according to Romans 4:5. [ibid.]
But when he does this, he does not work contrary to human nature.
Now God moves everything in its own manner, just as we see that in natural things, what is heavy and what is light are moved differently, on account of their diverse natures. Hence He moves man to justice according to the condition of his human nature. But it is man's proper nature to have free-will. Hence in him who has the use of reason, God's motion to justice does not take place without a movement of the free-will; but He so infuses the gift of justifying grace that at the same time He moves the free-will to accept the gift of grace, in such as are capable of being moved thus. [ibid.; emphasis added]
It should go without saying (although I fear otherwise) that Aquinas' point about how God moves man depends not upon the particular examples – based upon a medieval understanding of physics – that he uses to establish his analogy. The point is that God does not work contrary to the nature of a thing when he moves it; he doesn't toss rocks up into the sky when gravity ought to be pulling them down to the ground. And in the same way, he doesn't save a man in violation of the man's free will.
Even so, as St. Thomas points out, even this act of the free will is occasioned by grace, so that once again we see there is no possible way in which one could say that the Catholic Gospel is works-based. No. It is founded solely upon grace.