Andrew Sinclair was born at Paisley, Renfrewshire, Scotland, on 13 April 1794, the son of John Sinclair, a weaver, and his wife, Agnes Renfrew. He never married. From 1814 to 1818 Sinclair studied medicine and surgery at the University of Glasgow, at L'Hôpital de la Charité in Paris, and at the University of Edinburgh where he qualified as a licentiate in 1818. In 1822 he joined the Royal Navy as an assistant surgeon, and for 10 years, from 1823, served on the Owen Glendower at the Cape of Good Hope and in the Mediterranean. During this period he collected botanical and zoological specimens which he sent to the British Museum.
After taking further lectures in medicine, Sinclair joined the Sulphur in 1835 as surgeon, and accompanied Captain William Beechey on his survey expedition to the Pacific coasts of North and South America. Until he was invalided home in 1839, he continued to pursue his interest in botany, sending specimens from California, Mexico, Central America and Brazil to the British Museum and the Royal Botanic Gardens, Kew. These collections established his reputation as a foremost collector. After recovering his health he began a brief period as a surgeon on convict ships to Australia. On one voyage in 1841 he visited the Bay of Islands, New Zealand, arriving on the Favorite on 24 October. There he joined the missionary William Colenso, and Joseph Dalton Hooker, assistant surgeon on James Clark Ross's Antarctic expedition which was in New Zealand at the time, on several local botanical expeditions. He subsequently presented to the British Museum a collection of shells and animals which included the shellfish pipi, crabs, spiders, dragonflies, cicadas, butterflies, corals and sponges. He returned to Scotland the following year.
In September 1843 Sinclair arrived in Tasmania as surgeon superintendent on the convict ship Asiatic. After signing off he intended to return to England. However, in Sydney he met Robert FitzRoy, the governor elect of New Zealand. The two struck an immediate rapport, and FitzRoy offered Sinclair a free passage to Auckland. They arrived there on 23 December 1843. On 6 January 1844, after much persuasion from FitzRoy, Sinclair reluctantly accepted the appointment of colonial secretary, and was also made a member of the Legislative Council on 8 January.
From 1844 until the establishment of responsible government in April 1856 Sinclair served as colonial secretary under Governors Robert FitzRoy and George Grey, Acting Governor Robert Wynyard, and Governor Thomas Gore Browne. He had wide discretionary powers, but showed no particular ability in dealing with the continuing struggle between the governors and settlers which dominated politics in early Auckland. He established a reputation for being 'honest, upright, scrupulous and laborious', however, and is also credited with choosing and training subordinates who became the nucleus of an efficient public service.
In Auckland Sinclair also devoted himself to business transactions and a variety of cultural pursuits. He was widely regarded as a shrewd businessman, to whom many, including FitzRoy and Grey, entrusted their investments. He was fond of literature, music and art, and through his travels and thirst for knowledge commanded a rich repertoire of stories which he loved to relate. A staunch Presbyterian, he was a founder of St Andrew's Church, Auckland, in 1847, and was also a founder of the Auckland Museum in 1852.
Although his political career was unremarkable, Sinclair is best remembered for his contribution to natural history. During his term of office as colonial secretary he spent much of his spare time collecting botanical specimens for Kew. After his retirement he visited Scotland and Europe, where he discussed a wide range of scientific matters with Charles Darwin, Thomas Huxley and Richard Owen. On 20 January 1857 he was elected a fellow of the Linnean Society of London. He returned to New Zealand in late 1858 to collect material for J. D. Hooker's Handbook of the New Zealand flora (1864–67). On 20 February 1861 Sinclair joined Julius Haast on what was to be his last, and fateful, expedition – Haast's geological survey of the headwaters of the Rangitātā River, Canterbury. On 26 March 1861 Sinclair was drowned while crossing the Rangitātā. He was buried at Mesopotamia station nearby.
J. D. Hooker had dedicated his Flora Novae-Zelandiae (1853) to Andrew Sinclair and to two other prominent collectors, William Colenso and David Lyall. Sixteen New Zealand plants were named in Sinclair's honour, including an orchid, sedges, herbs, shrubs and trees, notably the puka ( Meryta sinclairii). Sinclair's efforts during the Beechey expedition were commemorated by W. J. Hooker and G. A. W. Arnott in the plant genus Sinclairia (Asteraceae), while his association with Haast is remembered in the mountain daisy Haastia sinclairii. Haast in turn honoured his friend and field companion by naming Mt Sinclair, near Mesopotamia, and the Sinclair River. Had he not died tragically, Sinclair might well have extended his collecting to rank equal with that of Colenso. On his death J. D. Hooker wrote: 'His loss has been a very great one, whether as a botanist or as an enthusiastic and liberal patron of science.'