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.
