Laura Trott, Joanna Rowsell Shand, Katie Archibald and Elinor Barker rode to an emphatic victory over Russia in a time of four minutes 17.010 seconds.
GB's male quartet of Sir Bradley Wiggins, Owain Doull, Andy Tennant and Jon Dibben sealed a memorable double by beating hosts Switzerland.
They clocked a time of three minutes 55.243 secs at the Velodrome Suisse.
Britain have won team pursuit gold in every European Championships they have entered since 2010.
The event in Grenchen marks Wiggins's first 2016 Olympics qualifying outing since he decided to return to the track from road racing.
Points earned will help towards securing places at next summer's Olympic Games in Brazil.
Great Britain's men and women won gold in their team pursuit finals at the European Track Championships.