There isn’t an equivalent with pip.
Best way is to pip install package && pip freeze > requirements.txt
You can see all the available options on their documentation page.
If it really bothers you, it wouldn’t be too difficult to write a custom bash script (pips) that takes a -s argument and freezes to your requirements.txt file automatically.
Edit 1
Since writing this there has been no change in providing an auto --save-dev option similar to NPM however Kenneth Reitz (author of requests and many more) has released some more info about a better pip workflow to better handle pip updates.
Edit 2
Linked from the “better pip workflow” article above it is now recommended to use pipenv to manage requirements and virtual environments. Having used this a lot recently I would like to summarise how simple the transition is:
Install pipenv (on Mac)
brew install pipenv
pipenv creates and manages it’s own virtual environments so in a project with an existing requirements.txt, installing all requirements (I use Python3.7 but you can remove the --three if you do not) is as simple as:
pipenv --three install
Activating the virtualenv to run commands is also easy
pipenv shell
Installing requirements will automatically update the Pipfile and Pipfile.lock
pipenv install <package>
It’s also possible to update out-of-date packages
pipenv update
I highly recommend checking it out especially if coming from a npm background as it has a similar feel to package.json and package-lock.json