Heinrich was born in Gangloffsömmern the son of Johann Moritz von Brühl, a noble who held the office of the Oberhofmarschall at the court of Saxe - Weissenfels, ruled by a cadet branch of the Albertine House of Wettin. His father was ruined and compelled to part with his family estate, which passed into the hands of the prince. Under Duke Christian of Saxe - Weissenfels von Brühl was first placed as page with the dowager duchess, and was then received at her recommendation into the court of Electoral Saxony at Dresden as a Silberpage on April 16, 1719. He rapidly acquired the favour of the Elector Frederick Augustus I of Wettin, surnamed the Strong, who had been elected King of Poland (as Augustus II) in 1697. Brühl was largely employed in procuring money for his extravagant master. He became Chief Receiver of Taxes and Minister of the Interior of Saxony in 1731.

He was at Warsaw when his master died in 1733, and obtained the confidence of the Electoral Prince Frederick Augustus II, who was at Dresden at the time, by acquiring the papers and jewels of his late father and bringing them promptly to his successor. Von Brühl raised money to secure the election of Frederick Augustus II as Polish king (Augustus III the Saxon), who in the following War of the Polish Succession prevailed against his rival Stanisław Leszczyński, though only with the armed support of Russia and the Habsburg Monarchy.

During most of the thirty years of the ineffective reign of August III the Saxon, he was the major confidant of the king and the de facto head of the Saxon court. For a time he had to acquiesce to the influence of old servants of the electoral house, but after 1738 he was in effect sole minister, a position for which he actually had neither the skills nor the knowledge. The title of a Prime Minister was created for him in 1746, but his power extended beyond that office as a classic court favourite. Besides securing huge grants of land, his titles spread over several lines of print, and he drew the combined salaries of these offices. He also worked closely with Bishop Kajetan Sołtyk of Kraków.

Brühl was a capable diplomat, who played a vital role in the Diplomatic Revolution of 1756 and the convergence of Habsburg and France. He must however be held wholly responsible for a ruinous fiscal policy which decisively weakened the position of Saxony within the Holy Roman Empire between 1733 and 1763; for the mistaken ambition which led Frederick Augustus II to become a candidate for the throne of Poland, which led to a civil war and did sustainable damage to the Polish sovereignty; for the engagements into which he entered in order to secure the support of Emperor Charles VI of Habsburg; for the shameless and ill-timed tergiversations of Saxony during the War of the Austrian Succession; for the intrigues which entangled the Electorate in the alliance against King Frederick II of Prussia, which led to the outbreak of the Seven Years' War; and for the waste and want of foresight which left the bankrupt country utterly unprepared to resist the immediate attack of the Prussian king. After a few weeks the decimated Saxon army under Frederick Augustus Rutowsky had to capitulate and was dissolved, while Saxony remained a war theatre.

He was not only without political or military capacity, but was so garrulous that he could not keep a secret. His indiscretion was repeatedly responsible for the king of Prussia's discoveries of the plans laid against him. Nothing could shake the confidence of his master, which survived the ignominious flight into Bohemia, into which he was trapped by von Brühl at the time of the Battle of Kesseldorf, and all the miseries of the Seven Years' War. The favourite abused the confidence of his master shamelessly. Not content with the 67,000 talers a month which he drew as salary for his innumerable offices, he was found when an inquiry was held in the next reign to have abstracted more than five million talers of public money for his private use. He left the work of the government offices to be done by his lackeys, whom he did not even supervise.

Von Brühl died at Dresden on 28 October 1763, having survived his master only for a few weeks. The new elector, Frederick Christian caused an inquiry to be held into his administration. His fortune including large palaces at Pförten (present-day Brody), Oberlichtenau and Wachau - Seifersdorf was found to amount to a million and a half talers, and was sequestered but afterwards restored to his family. In 1736 the architect Johann Christoph Knöffel had begun to build a city palace and terrace for the count on the bank of the Elbe in the heart of Dresden. This was originally called "Brühl's Garden" and is today known as Brühl's Terrace. The Brühl Palace at Warsaw was rebuilt according to the designs by Joachim Daniel von Jauch from 1754 to 1759. Von Brühl was a dedicated collector and protector of the arts - Francesco Algarotti called him a Maecenas. He owned a large gallery of pictures, which was bought by Empress Catherine II of Russia in 1768, and his library of 70,000 volumes was one of the biggest private libraries in the Holy Roman Empire.

In 1736 he had received the title of a Saxon Reichsgraf and married the countess Franziska von Kolowrat - Krakowska, a favourite of the wife of Frederick Augustus. Four sons and a daughter survived him. Alois Friedrich von Brühl (1739 – 1793) was also a Saxon politician and military commander. His youngest son, Hans Moritz von Brühl (d. 1811), was before the Revolution of 1789 a colonel in the French service, and afterwards general inspector of roads in Brandenburg and Pomerania. By his wife Margarethe Schleierweber, the daughter of a French corporal, but renowned for her beauty and intellectual gifts, he was the father of Karl Friedrich Moritz Paul von Brühl (1772 – 1837), a friend of Johann Wolfgang von Goethe, who as intendant general of the Prussian royal theatres was of some importance in the history of the development of the drama in Germany. In 1830 he was appointed intendant general of the royal museums. Another granddaughter was Marie von Brühl, who married Carl von Clausewitz.