Stratton acted as a producer on Newsnight before joining The Guardian. She also previously worked on Andrew Marr's talkshow on BBC Four and This Week.
Her return to Newsnight as political editor follows the departure of Crick, who joined Channel 4 News in July as the broadcaster's political correspondent.
"It's a privilege to be returning to Newsnight - it is a great programme with an important role in stimulating political debate, especially at such a turbulent time both here and abroad," said Stratton.
"I'm very sad, though, to be leaving The Guardian. I'm proud to have worked there and have benefited immensely from the wisdom, kindness and good humour of my colleagues."
Newsnight editor Peter Rippon added: "I am delighted to be able to welcome such a talented journalist as Allegra to the team. The Newsnight political editor is a cornerstone of the programme and Allegra's appointment is crucial to our continuing success.
"Our audiences expect us to hold Westminster to account and to bring them the inside stories, and with Allegra's experience I have every confidence she will be able to fulfil their expectations."
The BBC said that Stratton's start date on Newsnight, which is aired every weeknight on BBC Two at 10.30pm, will be "announced in due course".