Speaking in Waitangi on Sunday, the Wales star's thoughts were with the six people killed and at least 48 others injured in the incident at London Bridge and the Borough Market.
Warburton, whose team opened their tour with a 13-7 win over the New Zealand Provincial Barbarians on Saturday, said the Lions wanted to bring some joy to those affected.
"In future performances and wins, we'll dedicate to those who have been involved and I guess we can play a small part in trying to cheer a majority of the nation up by trying to be successful over here," he said.
"We'll commit our performances and all our efforts to those involved recently in London."
He added: "I just wanted to wish those involved all the best wishes from the Lions players and management.
"Our thoughts are always with the people back home in Britain and Ireland so thoughts and deepest sympathies go out to the families and people involved in that incident."
Our thoughts and deepest sympathies go out to those affected back home by the tragic events in London. pic.twitter.com/jNjpQ7vJYy
— British&Irish Lions (@lionsofficial) June 4, 2017
The Lions have nine matches left on their tour, including a three-Test series against the All Blacks.
Tour manager John Spencer also sent his regards, saying: "We're four nations, but we're one jersey, one team.
"We'll all stand united here about the sentiments in London."
The Lions are next in action against Super Rugby outfit Blues at Eden Park on Wednesday.