Milne was the son of distinguished admiral, Sir Alexander Milne, 1st Baronet, and grandson of Vice Admiral Sir David Milne.
For ten years he served on the royal yachts, commanding H.M. Yachts from 1903 to 1905 as Flag Officer, and being made a Rear-Admiral in 1904. During this time, he became a friend of King Edward VII and of Queen Alexandra, who called him "Arky-Barky".
His next posting was as second in command of the Atlantic Fleet until 1906. From 1908 to 1910 Milne commanded the 2nd Division of the Home Fleets. Milne was made a full Admiral in 1911 and given command of the Mediterranean Fleet in November 1912. The appointment was made by Winston Churchill, First Lord of the Admiralty, taking into consideration the views of the King, George V. Admiral Fisher, former First Sea Lord deplored Milne's appointment to such an important post, accusing Churchill of having betrayed the navy. At the outset of World War I, the Fleet was twenty-seven ships strong.
The German ships steamed out of Messina harbour at midnight, August 5, precisely as Britain officially went to war with Germany. They were headed for Turkey, to attempt to convince it to enter the war on Germany's side, by force if necessary. The heading surprised Admiral Milne who had expected them to steam west to the Straits of Gibraltar. He had only one ship, the light cruiser HMS Gloucester, in a position to follow them. The next morning (August 7), the Gloucester closed in and opened fire on the Breslau, which returned fire. Breslau was slightly damaged in the exchange receiving one hit at the waterline. Near the western coast of Greece, the pursuit of the Goeben and the Breslau was taken up by four more British ships, led by Milne's second-in-command, Rear-Admiral Sir Ernest Charles Thomas Troubridge. Troubridge's ships were smaller and slower than the Goeben, they were also outgunned. Troubridge and his gunnery officer determined they could not intercept the German ships before daylight. They concluded that the enemy battlecruiser's superior speed and range would allow it to maintain enough distance to pick off Toubridge's ships at leisure before they could ever get close enough to engage effectively.
Souchon's ships made it to Constantinople and were admitted into the harbour by the Turks. The German diplomats reminded the Turks that Great Britain had recently broken a contract to supply two new battleships to the Turkish government (which the British Admiralty had decided to keep for its own use as war loomed), and offered to sell them the Goeben and the Breslau. The Turks agreed on August 16 and eventually joined Germany's side on October 30, 1914. The ships were renamed the Yavuz Sultan Selim and the Midili, retaining their German crews; Souchon was made commander-in-chief of the Turkish Navy.
Afterwards, Milne served out the rest of the war on half-pay. He was offered the three-year command at the Nore encompassing the ports of Chatham and Sheerness in 1916, but the position eventually went to another officer due to "other exigencies". The Admiralty repeatedly emphasised that Milne had been exonerated of all blame in the affair, most significantly when announcing Milne's retirement at his own request in 1919, so as to further the promotion of other officers. In 1920 the official naval history of the war by Sir Julian Corbett was critical of Milne's handling of the affair and contained "serious inaccuracies". Milne requested the Admiralty to act which they declined to do so, and in 1921 Milne wrote The Flight of the Goeben and the Breslau in an attempt to clear his name, which "justified the official approbation".