OMD and Havas Media previously shared the account, with OMD handling media buying and Havas Media working on media planning. Havas was knocked out earlier in the pitch process.
Camelot began the review in March, in reaction to what it called the "increasingly complex and fragmented" media landscape.
Camelot operates brands including Lotto, EuroMillions and National Lottery Instants.
Sally Cowdry, the marketing and consumer director of Camelot, said: "Today’s increasingly fragmented and complex media landscape has led to a significant shift of consumer behaviour.
"We were highly impressed with the quality of work submitted by all agencies throughout the pitch process and would like to thank them all for their hard work.
"Vizeum has a fantastic track record across both the traditional and digital media landscapes, and presented some truly unique and exciting ideas. We are very much looking forward to working with them on our new integrated media brief."
Oystercatchers managed the review process.
Richard Morris, the managing director of Vizeum, said: "We are, of course, thrilled to be partnering with Camelot.
"The National Lottery makes millionaires and funds National Lottery projects, changing lives on a personal, local and national level – we are looking forward to helping bring these stories to life.
"This creates a massive opportunity for media to demonstrate the power of The National Lottery through a digitally-integrated approach."