The BBC One series, in which Suranne Jones plays a GP who suspects her husband of infidelity, beat BBC Two's Wolf Hall and Channel 4's Humans.
Other honourees include British film 45 Years, grime artist Stormzy and Mercury music prize winner Benjamin Clementine.
Elusive artist Banksy missed out on the visual art award for his Dismaland theme park in Weston-super-Mare.
That prize went instead to painter Lynette Yiadom-Boakye for her Verses After Dark exhibition at the Serpentine in London.
Martin McDonagh's black comedy Hangmen was the recipient of the theatre award, while Rob Delaney and Sharon Horgan received the comedy award for their Channel 4 sitcom Catastrophe.
Sunday's ceremony at the Savoy Hotel in London also saw Eddie Izzard receive an outstanding contribution award for his contribution to British arts.
The fundraiser and campaigner said it was "a great honour", albeit "bonkers", to get the accolade from fellow comedian Sir Lenny Henry.
Stormzy - real name Michael Omari - received a Breakthrough award at the event, which was hosted by Lord Melvyn Bragg and will be broadcast on Sky Arts on 8 June.
The ceremony also saw Australian comedian Tim Minchin perform a song from his upcoming musical version of the 1993 film Groundhog Day.
Now celebrating their 20th anniversary, the awards are presented annually to "the very best of British culture and achievement".
FULL LIST OF WINNERS
Also nominated - Stephen Hough, International Piano Series: Debussy and Chopin, Royal Festival Hall; City of Birmingham Symphony Orchestra, Andris Nelsons' Farewell Concert, Symphony Hall
Also nominated - Brooklyn, Ex Machina
Also nominated - Chewing Gum, E4; Peter Kay's Car Share, BBC One
Also nominated - Paradise Lost (lies unopened beside me), Lost Dog; Woolf Works, Royal Ballet
Also nominated - The Past, Tessa Hadley; The Wolf Border, Sarah Hall
Also nominated - Krol Roger, Royal Opera House; Saul, Glyndebourne
Also nominated - Sleaford Mods, Key Markets; Years and Years, Communion
Also nominated - Oresteia, Almeida Theatre; People, Places And Things, National Theatre and Headlong
Also nominated - Humans, Channel 4; Wolf Hall, BBC Two
Also nominated - Banksy: Dismaland Bemusement Park; Cornelia Parker: Magna Carta (An Embroidery), British Library
Doctor Foster has been honoured at this year's South Bank Sky Arts Awards, winning the prize for best TV drama.