UPDATE 2 Since Git 2.5.0 the feature described below can be enabled on server side with configuration variable uploadpack.allowReachableSHA1InWant, here the GitHub feature request and the GitHub commit enabling this feature. Note that some Git servers activate this option by default, e.g. Bitbucket Server enabled it since version 5.5+. See this answer on Stackexchange for a exmple of how to activate the configuration option.

UPDATE 1 For Git versions 1.7 < v < 2.5 use git clone and git reset, as described in Vaibhav Bajpai's answer

If you don't want to fetch the full repository then you probably shouldn't be using clone. You can always just use fetch to choose the branch that you want to fetch. I'm not an hg expert so I don't know the details of -r but in git you can do something like this.

# make a new blank repository in the current directory
git init

# add a remote
git remote add origin url://to/source/repository

# fetch a commit (or branch or tag) of interest
# Note: the full history up to this commit will be retrieved unless 
#       you limit it with '--depth=...' or '--shallow-since=...'
git fetch origin <sha1-of-commit-of-interest>

# reset this repository's master branch to the commit of interest
git reset --hard FETCH_HEAD

No need to download the whole history, and no need to call git init:

git clone --depth=1 URL
git fetch --depth=1 origin SHA1
git checkout SHA1
git branch -D @{-1}  # if you want to tidy up the fetched branch

This has the disadvantage, to CB Baileys answer, that you will still download 1 unnecessary revision. But it's technically a git clone (which the OP wants), and it does not force you to download the whole history of some branch.