Sweelinck was born in Deventer, Netherlands, in April or May 1562. He was the eldest son oforganist Peter (or Pieter) Swybbertszoon and Elske Jansdochter Sweeling, daughter of a surgeon. Soon after Sweelinck's birth, the family moved to Amsterdam, where from about 1564, Pieter Swybbertszoon served as organist of the Oude Kerk (Sweelinck's paternal grandfather and uncle also were organists). Jan Pieterszoon must have received first lessons in music from his father. Unfortunately, his father died in 1573. He subsequently received general education under Jacob Buyck, Catholic pastor of the Oude Kerk (these lessons stopped in 1578 after the Reformation of Amsterdam and the subsequent conversion to Calvinism; Buyck chose to leave the city). Little is known about his music education after the death of his father; his music teachers may have included Jan Willemszoon Lossy, a little-known countertenor and shawm player at Haarlem, and/or Cornelis Boskoop, Sweelinck's father's successor at the Oude Kerk. If Sweelinck indeed studied in Haarlem, he was probably influenced to some degree by the organists of St.-Bavokerk, Claas Albrechtszoon van Wieringen and Floris van Adrichem, both of whom improvised daily in the Bavokerk.
According to Cornelis Plemp, a pupil and friend of Sweelinck's, he started his 44-year career as organist of the Oude Kerk in 1577, when he was 15. This date, however, is uncertain, because the church records from 1577–80 are missing and Sweelinck can only be traced in Oude Kerk from 1580 onwards; he occupied the post for the rest of his life. Sweelinck's widowed mother died in 1585, and Jan Pieterszoon took responsibility for his younger brother and sister. His salary of 100 florins was doubled the next year, presumably to help matters. In addition, he was offered an additional 100 guilders in the event that he married, which happened in 1590 when he married Claesgen Dircxdochter Puyner from Medemblik. He was also offered the choice between a further 100 guilders and free accommodations in a house belonging to the town, the latter of which he chose. Sweelinck's first published works date from around 1592–94: three volumes of chansons, the last of which is the only remaining volume published in 1594 (for reasons that are not certain, the composer adopted his mother's last name; "Sweelinck" first appears on the title-page of the 1594 publication). Sweelinck then set to publishing psalm settings, aiming to set the entire Psalter. These works appeared in four large volumes published in 1604, 1613, 1614 and 1621. The last volume was published posthumously and, presumably, in unfinished form. Sweelinck died of unknown causes on 16 October 1621 and was buried in the Oude Kerk. He was survived by his wife and five of their six children; the eldest of them, Dirck Janszoon, succeeded his father as organist of the Oude Kerk.
The composer most probably spent his entire life in Amsterdam, only occasionally visiting other cities in connection with his professional activities: he was asked to inspect organs, give opinions and advice on organ building and restoration, etc. These duties resulted in short visits to Delft,Dordrecht (1614), Enkhuizen, Haarlem (1594), Harderwijk (1608), Middelburg (1603), Nijmegen (1605), Rotterdam (1610), Rhenen (1616), as well as Deventer, his birthplace (1595, 1616). Sweelinck's longest voyage was to Antwerpen in 1604, when he was commissioned by the Amsterdam authorities to buy a harpsichord for the city. No documentary evidence has turned up to support the tradition, going back to Mattheson, that Sweelinck visited Venice – perhaps a confusion with his brother, the painter Gerrit Pietersz Sweelink, who did – and similarly there is no evidence that he ever crossed the English Channel, although copies of his music did such as the pieces included in the Fitzwilliam Virginal Book. His popularity as a composer, performer and teacher increased steadily during his lifetime. Contemporaries nicknamed him Orpheus of Amsterdamand even the city authorities frequently brought important visitors to hear Sweelinck's improvisations.