The France international's arrival from Real Sociedad was announced on Monday, with the transfer set to be completed subject to a medical.
Griezmann scored 16 goals in 35 La Liga outings for Sociedad in 2013-14, with his performances enough to earn a call-up to Didier Deschamps' squad for the World Cup in Brazil.
Following the departure of Diego Costa to Chelsea, Atletico have now captured Griezmann and Mario Mandzukic ahead of the defence of their Liga crown.
And Simeone believes Griezmann's speed can help the team in their bid to achieve that goal.
"He can give us solutions and he has characteristics which, until now, the team didn't have," said the Argentinian.
"A few of our pacier, quick-thinking players like Diego Costa, Adrian [Lopez] and [David] Villa have left and we need to replace them with players in a similar mould.
"The technical department have been working through all of this during the past 40 days.
"Up front we've lost Costa and Mandzukic has arrived. Griezmann will give us the velocity we've lacked and needed.
"You don't confect a team from just 11 players, but 18 or 20."