Yes, it might take a couple of weeks, but the backlinks will be forwarded to the new site. The total move might take a couple of months because pages will stay in googles index for a little while
I did this same exact thing to one of my sites. I went from a 17 letter domain name, to a 5 letter. So I set up a 301 redirect, and in a matter of days google started redirecting pages, links - everything to the new site.
Its been 2 months since I made the change and just about all of the links and everything have been moved in the search engines. If you do a google search for my old domain name, the new name ranks in 1st place. The old domain is not even listed. If you do a search for site: old domain, there are only 125 cached pages left out of several thousand.
If I do a link: old domain, there are still a lot of links that have not been updated. But google does not update the links but every few weeks anyway.