Swynnerton was born in Kersal, near Manchester, one of seven daughters of Francis Robinson, a solicitor. From an early age she painted watercolours to supplement the family's reduced income, but began her serious training as an artist at Manchester School of Art, before leaving to enrol at the Académie Julian in Paris. Her work was first exhibited at the Royal Academy in 1879, and the following year she exhibited a portrait of her friend Isabel Dacre (Manchester City Art Gallery), with whom she later formed the Manchester Society of Women Painters.
Swynnerton completed her studies by travelling for two years in Italy. During a stay in Rome she met the Manx sculptor Joseph Swynnerton, whom she married in 1883; until his death in 1910, they lived mainly in Rome. Whilst in Italy, Swynnerton painted works such as An Italian Mother and Child (Manchseter City Art Gallery) in a style clearly reminscent of Renaissance painting, and panoramic landscapes such as The Olive Gatherers (Manchester City Art Gallery).
In 1902, after a gap of sixteen years, Swynnerton exhibited again at the Royal Academy. Always greatly admired by other painters, her work was bought by prominent figures in the art world. In 1906 Sir George Clausen purchased New-Risen Hope, depicting the figure of a naked child, and later presented it to the National Gallery of Victoria in Melbourne. John Singer Sargent bought The Oreads in 1907, a sculpturesque group of sea-nymphs, giving the painting to the Tate Gallery, London, in 1922.
In addition to her allegorical paintings, Swynnerton exhibited many portraits at the Academy in the 1910s. In 1922, backed by Clausen and Sargent, Swynnerton was the first woman to be elected an Associate of the Royal Academy.