Brennan moves to Melbourne, Australia in 1861 with his parents. He starts his career as a watchmaker and a few years later is articled to Alexander Kennedy Smith, a renowned civil and mechanical engineer of the period. He serves as a sergeant in the Victorian Engineers under the command of Captain John James Clark. He invents the idea of a steerable torpedo in 1874, from observing that if a thread is pulled on a reel at an angle with suitable leverage, the reel will move away from the thread side. He spends some years working out his invention, and receives a grant of £700 from the Victorian government towards his expenses. He patents the Brennan torpedo in 1877. The idea is trialed at Camden Fort near Crosshaven, County Cork.
Brennan goes to England in 1880 and brings his invention before the War Office. Sir Andrew Clarke alerts the authorities to the possibilities of the torpedo if used in the defence of harbours and channels, and the patent is eventually bought for a sum believed to be more than £100,000 (£ 9,331,100 in 2019). In 1887 he is appointed superintendent of the Brennan torpedo factory, and is consulting engineer from 1896 to 1907.
Brennan does much work on a gyro monorail locomotive which is kept upright by a gyrostat. In 1903 he patents a gyroscopically-balanced monorail system that he designs for military use. He successfully demonstrates the system on November 10, 1909, at Gillingham, England, but fears that the gyroscopes might fail prevents adoption of the system for widespread use.
From 1916 to 1919 Brennan serves in the munitions inventions department. From 1919 to 1926 he is engaged by the air ministry in aircraft research work at the Royal Aircraft Establishment, Farnborough, and gives much time to the invention of a helicopter. The government spends a large sum of money on it, but in 1926 the air ministry gives up working on it, much to Brennan’s disappointment.
Brennan marries Anna Quinn on 10 September 10, 1892. The marriage results in a son and a daughter. He is created a Companion of the Order of the Bath in 1892, and is foundation member of the National Academy of Ireland in 1922.
In January 1932 Brennan is knocked down by a car at Montreux, Switzerland, and dies on January 17, 1932. He is buried at St. Mary’s Catholic Cemetery, Kensal Green, London, in an unmarked plot numbered 2454 that is opposite the Chapel record office. On March 11, 2014, Taoiseach Enda Kenny unveils a new gravestone for Brennan at St. Mary’s in a ceremony honouring the inventor’s life and career.
Gillingham Library retains the archive of his papers.