Pakenham, the third son of Thomas Pakenham, 1st Baron Longford (1713–1766), entered the Royal Navy in 1771 on board the Southampton, with Captain John MacBride, with whom he moved to the Orpheus in 1773. In 1774 he was on the coast of Guinea with William Cornwallis in the Pallas, and in 1775 was acting lieutenant of the Sphinx on the coast of North America.
In the following year he was promoted by Lord Shuldham to be lieutenant of the frigate Greyhound, and while in her saw much boat service, in the course of which he was severely wounded. In 1778 he joined the Courageux, commanded by Lord Mulgrave, in the fleet under Keppel, and was present in the Battle of Ushant on 27 July.
In the following spring he was moved into the Europe, going to North America with the flag of Rear-Admiral Mariot Arbuthnot, and on 21 September 1779 was promoted to the command of the sloop Victor, newly captured from the enemy. He was then sent to the Jamaica station, where, on 2 March 1780, he was posted by Sir Peter Parker the elder to the San Carlos. His old wound, received while in the Greyhound, broke out again, and compelled him to return to England in the autumn.
In December 1780 he was appointed to the Crescent of 28 guns, attached to the fleet under George Darby, which relieved Gibraltar in April 1781, and was sent on to Menorca in company with the Flora under William Peere Williams-Freeman. On their way back, in passing through the straits, they fell in, on 30 May, with two Dutch frigates. In the ensuing Battle of Cape St Mary, one of the Dutch frigates, the Castor (commanded by Pieter Melvill van Carnbee), struck to the Flora, while the other, the Den Briel, overpowered and captured the Crescent. The Crescent was immediately recaptured by the Flora, the Den Briel making her escape; but both Crescent and Castor had received so much damage in the action that they fell into the hands of two French frigates on the way home, 19 June, the Flora escaping. Pakenham had, however, refused to resume the command of the Crescent, maintaining that by his surrender to the Den Briel his commission was cancelled, and that when recaptured the ship was on the same footing as any other prize.
For the loss of his ship he was tried by court-martial and honourably acquitted, it being proved that he did not strike the flag till, by the fall of her masts and the disabling of her guns, further resistance was impossible. He was therefore at once appointed to the frigate Minerva, which he commanded in the following year at the relief of Gibraltar by Lord Howe.
In 1793 he commissioned the Invincible, and in her took part in the Glorious First of June, when his conduct was spoken of as particularly brilliant, and he was recommended by Howe for the gold medal. In 1795 he was turned over to the 84-gun ship Juste, in the capture of which, on 1 June, he had had a principal hand. He was afterwards for some time master-general of the ordnance in Ireland, and had no further service in the navy.
In 1783, Pakenham entered the Irish House of Commons for Longford Borough and sat until 1790. Subsequently, he represented Kells until 1798 and again Longford Borough until the Act of Union in 1801.
On 14 February 1799, Pakenham was promoted to be rear-admiral, vice-admiral on 23 April 1804, and admiral on 31 July 1810. He was appointed a Knight Grand Cross of the Order of the Bath on 20 May 1820, and died on 2 February 1836. He married in 1785 Louisa, daughter of the Right Hon. John Staples, and had a large family. His fifth son Sir Richard Pakenham (1797–1868) was a diplomat who served as British ambassador to Mexico, the United States and Portugal.