As noted in another answer, there is now a command for this:
nvm now has a command to update npm. It’s
nvm install-latest-npmornvm install --latest-npm.
nvm install-latest-npm: Attempt to upgrade to the latest working npm on the current Node.js version.
nvm install --latest-npm: After installing, attempt to upgrade to the latest working npm on the given Node.js version.
Below are previous revisions of the correct answer to this question.
For later versions of npm it is much simpler now. Just update the version that nvm installed, which lives in ~/.nvm/versions/node/[your-version]/lib/node_modules/npm.
I installed Node.js 4.2.2, which comes with npm 2.14.7, but I want to use npm 3. So I did:
cd ~/.nvm/versions/node/v4.2.2/lib
npm install npm
Easy!
And yes, this should work for any module, not just npm, that you want to be “global” for a specific version of node.
In a newer version, npm -g is smart and installs modules into the path above instead of the system global path.