Vince Cable has hailed his Liberal Democrats as “the big winners” of the local election after the party gained hundreds of seats and control of a series of councils.

With all the results in, the Lib Dems had 1,350 councillors, a net gain of 703 on 2015 figures, and controlled 18 councils, in an election expected to be Cable’s swansong as leader. However, the gains came in the context of results that were disastrous for the party the last time the same areas were fought.

Among the areas taken were the traditional Tory stronghold of Cotswold district council, where the Lib Dems overtook the Conservatives to become the biggest single party, winning 18 of the 34 seats.

Arguably more striking was Bath and North East Somerset, where the party gained 23 seats to easily overturn the previous Tory majority, winning 37 of the 59 seats – including the home ward of the Tory Brexiter MP Jacob Rees-Mogg.

As the results came in, Cable travelled to Chelmsford in Essex, where his party overturned a previous 45-seat Conservative majority, amassing a net gain of 26 seats, while the Tories lost 31.

The party also gained Winchester, Vale of White Horse, Mole Valley, Hinckley and Bosworth, North Devon and North Norfolk. They also took the newly formed authority of Somerset West and Taunton, formed by the merger of two councils.

Cable told celebrating Lib Dem activists that the result reflected a “story across the country”. He said: “The Lib Dems were written off at one point but we’re coming back very, very strongly. We’re the big winners of the night throughout the country.”

In an earlier comment, Cable – who has said he will step down as leader following the local elections – said the Lib Dems had chalked up “already our best result for over 15 years”.

He said: “Voters have sent a clear message that they no longer have confidence in the Conservatives, but they are also refusing to reward Labour while the party prevaricates on the big issue of the day: Brexit.”

Cable added: “Today’s Lib Dem surge is a springboard to the European elections, where we are clearly the strongest remain force on the ballot paper. Every Liberal Democrat vote on 23 May will be a vote to stop Brexit.”

However, the result seemed to have a broader impetus than just Brexit, with analysis showing the party only did slightly better in remain-voting areas than those that supported leave in 2016. Chelmsford, perhaps the most dramatic Lib Dem gain, supported leave in the referendum.

The Lib Dems had been widely expected to gain seats, as the last time the same councils were fought for was May 2015, when the party was being roundly punished by voters for the Tory-Lib-Dem coalition.

Then, the Lib Dems lost more than 400 seats and four councils. In the general election on the same day the party slumped from 57 Westminster seats to eight.

The party’s home affairs spokesman, Ed Davey MP, said that nonetheless Thursday’s results marked a “sea change in British politics”, and were as good as the Lib Dems had ever seen – for example following the 2003 Iraq war.

“It is just a fabulous night. We are clearly back in the game,” he told BBC Radio 4’s Today programme. “People have been frustrated with the appalling Tory government, who have let them down not just on Brexit but with cuts to police and schools, and a split opposition with such poor leadership.

“They have been crying out for a strong alternative. The Liberal Democrats have proven we are that strong alternative to the Tories and Labour.”