In addition two paragraphs later the narrator tells us this

Margaret, the other sister, was a good-humoured, well-disposed girl; but as she had already imbibed a good deal of Marianne's romance, without having much of her sense;

With the title as it is, Austen must have meant to use those specific words to describe Marianne for a reason.

I admit I did get my definitions from a current 2007 Oxford Dictionary and Thesaurus. The only definitions of "sensible" regarding "feeling" were - b perceptible by the senses c great enough to be perceived.
Neither of those uses of sensible as adj. could be applied to a person.

I have read of the idea of "sensibility" as BarbaraB says above. It is not something I've studied- just by reading the critiques contained in my copies of the book S&S.

Here are a few quotes from the Henry Hitchings afterword in my B&N Collectors library edition from 2003
The words 'sense' and 'sensibility' were highly charged at the turn of the century. 'Sensibility' meant something akin to 'fine feeling' or 'delicacy of perception'. 'Sense,' though it could certainly imply 'common sense' as it does today, would have connoted restraint and social responsibility.

But Austen has just told us that Marianne has sense. So does Marianne have common sense? restraint (snort) and social responsibility?

One more quote from Hitchings- speaking about 18th century new theory
Rousseau's influential doctrines laid considerable emphasis on the importance of feeling as a mainstay of identity: men and women were intrinsically good, but society corrupted them by distorting their urges and emotions. One of the indirect results of Rousseau's work was a rage for sentimental literature- for books about feelings and their role in creating a sense of self

Here are a couple of quote's from the introduction by Peter Conrad in my copy of 1992 Everyman's Library edition
Critics of Sense and Sensibility have doubly injured it, first by misreading it as a schematic and even allegorical exposition of the opposite qualities joined in its title, and then blaming it for an immature two-dimensionality inflicted on it by themselves ...
The critical fallacy is to insist that sense and sensibility are explicit ideological categories, when in truth Jane Austen is treating not ideas but their decomposition into modes of thinking and feeling...

Sense and Sensibility are not antagonistic but interdependent... 'Sense' generates 'sensibility' and ramifies into variant terms like 'sensitivity' or 'sentimentality' each extending and qualifying the original meaning.

I tend to agree with the idea that sense cannot be the opposite of sensibility, just because that makes more sense :-) to me.
I can also imagine Marianne reading all of that sentimental literature of the day and deciding that is the correct way to act.

Just one more quote from Peter Conrad to stew over...
...but in unfurling the complicated relations of the sisters it also reveals that the better part of Elinor's sense is a courageous and generous sensibility, and the better part of Marianne's sensibility is a prudent, even rigorous good sense.