Assistant Commissioner Neil Basu said: "This is shocking and tragic news.
"Dawn leaves behind her family, including three children, and our thoughts and prayers are with them at this extremely difficult time.
"The 45-year-old man who fell ill with Dawn remains critically ill in hospital and our thoughts are with him and his family as well.
"This terrible news has only served to strengthen our resolve to identify and bring to justice the person or persons responsible for what I can only describe as an outrageous, reckless and barbaric act.
"Detectives will continue with their painstaking and meticulous work to gather all the available evidence so that we can understand how two citizens came to be exposed with such a deadly substance that tragically cost Dawn her life."
Sturgess's family are being given support by specialist officers and have asked for privacy.
Prime Minister Theresa May said she was "appalled and shocked", and sent her "thoughts and condolences" to the family.
"Police and security officials are working urgently to establish the facts of this incident, which is now being investigated as a murder," she added.
"The Government is committed to providing full support to the local community as it deals with this tragedy."
A post-mortem examination will be scheduled to take place in due course.
Sajid Javid, the Home Secretary, had earlier said there were "no current plans" to impose fresh sanctions on Russia, despite accusing the country of using Britain as a "dumping ground" for poison.
During a brief tour of Amesbury today, he said he would not "jump to conclusions" over the latest attack.
It comes four months after former Russian spy Sergei Skripal and his daughter Yulia fell ill after being exposed to Novichok in nearby Salisbury.
Police have yet to identify any individuals responsible for the poisoning, and are working to establish how Sturgess and Rowley came into contact with the nerve agent.
Javid said: "Clearly what we have already determined, what our expert scientists have determined, is that the nerve agent in this incident is the exact same nerve agent as was used back in March (when the Skripals were poisoned).
"We know back in March that was the Russians. We know it was a barbaric inhuman act by the Russian state. Again for this particular incident we need to learn more and let the police do their work."