The blue-chip FTSE 100 index closed up 0.2%, while the domestically-focused mid-cap FTSE 250 ended 0.6% higher after opening lower. Gains were led by travel and leisure <.FTNMX5750>, banks <.FTNMX8350> and insurers <.FTNMX8570>.
Sunak announced new measures to help protect jobs, easing the eligibility for his jobs support scheme, reducing the cost for employers, and announcing new cash grants for affected businesses.
"Today's move (by Sunak) is a recognition of the economic impact of tier two restrictions and the fact that the COVID-19 situation in the UK is worsening rather than improving," said Edward Park, chief investment officer at Brooks Macdonald Asset Management.
AstraZeneca shares rose 0.4% after a study found AstraZeneca Plc Oxford COVID-19 vaccine accurately followed the genetic instructions programmed into it to successfully provoke a strong immune response.
Fears about new coronavirus-led restrictions as well as Brexit uncertainty have weighed on UK markets recently, with the FTSE 100 on course for its second straight week of declines.
Three more areas of England were put in the "high" category of the UK's three-tier restriction system on Thursday as infections accelerate swiftly.
Meanwhile, with time fast running out, Britain and the European Union started intensified negotiations on Thursday in a new, final push to try to secure a deal to protect billions of dollars of post-Brexit trade between the neighbours.
In company news, Unilever Plc rose 0.4% after it reported a stronger-than-expected return to sales growth in the third quarter.
Rentokil Initial Plc gained 3.9% as the pest control firm reported a 9.8% rise in third-quarter revenue due to higher demand for its disinfection services.
(Reporting by Devik Jain in Bengaluru and Joice Alves in London, editing by Uttaresh.V, Andrew Heavens and Susan Fenton)
By Devik Jain and Joice Alves