The Twickenham MP told BBC Breakfast that the gains of 676 after Thursday’s vote were “the best results we’ve had in the 40 years of our existence”.
Reflecting on several years of election losses, Sir Vince said the Lib Dems’ positioning as a Remain party will win them votes in the upcoming European elections.
He told the BBC: “We are clearly a major force, we are clearly the leading Remain party and we expect to do well on the basis of that. We have a long history of support for Europe.
“When people are trying to make their minds up, they would and they should vote for us, knowing that every vote is a vote to stop Brexit.”
The Lib Dems have taken control of ten councils, including Winchester and North Norfolk, which Sir Vince credited to “lots of hard work, over months”.
The party increased its presence across England, including in Chelmsford, where it gained 26 seats and took control from the Conservatives.
The town, which had a population of 168,310 according to the 2011 census, voted 53,249 in favour of Leave in 2016, compared with 47,545 to Remain.
Thursday’s poll saw the Conservatives lose more than 1,300 seats and 45 councils – the worst performance, in raw numbers, by a governing party in local elections since 1995.
Meanwhile, the number of councils under no overall control has increased by 36, to 71 in total.
Additional reporting by Press Association.