tell pip to install the dependencies of packages listed in a requirement file

Simply use:

pip install -r requirements.txt

This installs all dependencies listed in requirements.txt file.

Any way to do this without manually re-installing the packages in a new virtualenv to get their dependencies ? This would be error-prone and I'd like to automate the process of cleaning the virtualenv from no-longer-needed old dependencies.

That's what pip-tools package is for (from https://github.com/jazzband/pip-tools):

Installation

$ pip install --upgrade pip  # pip-tools needs pip==6.1 or higher (!)
$ pip install pip-tools

Example usage for pip-compile

Suppose you have a Flask project, and want to pin it for production. Write the following line to a file:

# requirements.in
Flask

Now, run pip-compile requirements.in:

$ pip-compile requirements.in
#
# This file is autogenerated by pip-compile
# Make changes in requirements.in, then run this to update:
#
#    pip-compile requirements.in
#
flask==0.10.1
itsdangerous==0.24        # via flask
jinja2==2.7.3             # via flask
markupsafe==0.23          # via jinja2
werkzeug==0.10.4          # via flask

And it will produce your requirements.txt, with all the Flask dependencies (and all underlying dependencies) pinned. Put this file under version control as well and periodically re-run pip-compile to update the packages.

Example usage for pip-sync

Now that you have a requirements.txt, you can use pip-sync to update your virtual env to reflect exactly what's in there. Note: this will install/upgrade/uninstall everything necessary to match the requirements.txt contents.

$ pip-sync
Uninstalling flake8-2.4.1:
  Successfully uninstalled flake8-2.4.1
Collecting click==4.1
  Downloading click-4.1-py2.py3-none-any.whl (62kB)
    100% |████████████████████████████████| 65kB 1.8MB/s
  Found existing installation: click 4.0
    Uninstalling click-4.0:
      Successfully uninstalled click-4.0
Successfully installed click-4.1