Is there a Wikipedia API just for retrieve the content summary?

I need just to retrieve the first paragraph of a Wikipedia page.

Content must be HTML formatted, ready to be displayed on my website (so no BBCode, or Wikipedia special code!)

There's a way to get the entire "introduction section" without any HTML parsing! Similar to AnthonyS's answer with an additional explaintext parameter, you can get the introduction section text in plain text.

Query

Getting Stack Overflow's introduction in plain text:

Using the page title:

https://en.wikipedia.org/w/api.php?format=json&action=query&prop=extracts&exintro&explaintext&redirects=1&titles=Stack%20Overflow

Or use pageids:

https://en.wikipedia.org/w/api.php?format=json&action=query&prop=extracts&exintro&explaintext&redirects=1&pageids=21721040

JSON Response

(warnings stripped)

{
    "query": {
        "pages": {
            "21721040": {
                "pageid": 21721040,
                "ns": 0,
                "title": "Stack Overflow",
                "extract": "Stack Overflow is a privately held website, the flagship site of the Stack Exchange Network, created in 2008 by Jeff Atwood and Joel Spolsky, as a more open alternative to earlier Q&A sites such as Experts Exchange. The name for the website was chosen by voting in April 2008 by readers of Coding Horror, Atwood's popular programming blog.\nIt features questions and answers on a wide range of topics in computer programming. The website serves as a platform for users to ask and answer questions, and, through membership and active participation, to vote questions and answers up or down and edit questions and answers in a fashion similar to a wiki or Digg. Users of Stack Overflow can earn reputation points and \"badges\"; for example, a person is awarded 10 reputation points for receiving an \"up\" vote on an answer given to a question, and can receive badges for their valued contributions, which represents a kind of gamification of the traditional Q&A site or forum. All user-generated content is licensed under a Creative Commons Attribute-ShareAlike license. Questions are closed in order to allow low quality questions to improve. Jeff Atwood stated in 2010 that duplicate questions are not seen as a problem but rather they constitute an advantage if such additional questions drive extra traffic to the site by multiplying relevant keyword hits in search engines.\nAs of April 2014, Stack Overflow has over 2,700,000 registered users and more than 7,100,000 questions. Based on the type of tags assigned to questions, the top eight most discussed topics on the site are: Java, JavaScript, C#, PHP, Android, jQuery, Python and HTML."
            }
        }
    }
}

Documentation: API: query/prop=extracts