zlib handles the deflate compression/decompression algorithm, but there is more than that in a ZIP file.
You can try libzip. It is free, portable and easy to use.
UPDATE: Here I attach quick'n'dirty example of libzip, with all the error controls ommited:
#include <zip.h>
int main()
{
//Open the ZIP archive
int err = 0;
zip *z = zip_open("foo.zip", 0, &err);
//Search for the file of given name
const char *name = "file.txt";
struct zip_stat st;
zip_stat_init(&st);
zip_stat(z, name, 0, &st);
//Alloc memory for its uncompressed contents
char *contents = new char[st.size];
//Read the compressed file
zip_file *f = zip_fopen(z, name, 0);
zip_fread(f, contents, st.size);
zip_fclose(f);
//And close the archive
zip_close(z);
//Do something with the contents
//delete allocated memory
delete[] contents;
}