Kidman was named best actress for her role as pioneering British scientist Rosalind Franklin in the play Photograph 51.
The award for best actor went to McAvoy for his performance in The Ruling Class.
The awards, hosted by Rob Brydon, took place at London’s Old Vic theatre on Sunday night.
Picking up her award, Kidman paid a moving tribute to her scientist father who died in 2014.
“Every night before I would go on stage I would kneel in front of his photograph and look into his eyes and he would give me the strength to go and do it every night,” she said.
The Oscar-winning star had returned to the London stage for the first time in 17 years to play Franklin, the only woman involved in the discovery of DNA’s double helix in 1953.
Among the stars attending the event and handing out awards were Gillian Anderson, Shirley Bassey, Salma Hayek, Gugu Mbatha-Raw, Dominic West and Ruth Wilson.
Imelda Staunton received a standing ovation as she picked up the best musical performance prize for Gypsy. She scooped the same category at last month’s UK Theatre Awards.
Written in the 1950s with lyrics by Stephen Sondheim, Gypsy opened in Chichester in 2014 before it transferred to London’s West End.
Staunton described Sondheim as the Shakespeare of musical theatre. “Every night when I sing his speeches – as I think of them – I revel in the richness, depth, pain and wit of his words,” she said.
Sondheim, who turned 85 this year, was honoured on stage with a special award presented by co-hosts Dame Judi Dench and Sir Ian McKellen at the end of the ceremony.