The Cardiff-based 19-year-old, along with Kathleen Dawson, Siobhan-Marie O'Connor and Fran Halsall, triumphed at the European Championships in London.
Tutton, part of Team GB's squad for the Rio Games, also won 100m breaststroke bronze at the championships.
"It's just been up and up. It's such a good season for me," she said.
"I keep going up and to finish on a high here, it's incredible."
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Tutton set a British record for the 200m breaststroke in April, which sealed her place at Rio.
Sharron Davies, the 1980 Olympic silver medallist, has said the City of Cardiff club swimmer's "meteoric rise" has been a boost for British swimming.
Tutton hopes Britain's 4x100m medley relay quartet can repeat the performance which saw them win gold at the London Aquatics Centre when they go to Brazil in August.
"Hopefully we can replicate the same in Rio," she said, adding that she had learned from her time with the British team.
"It's so nice to be around such experienced swimmers, just to see how they prepare themselves and just to be around such positive surroundings."
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Olympics-bound swimmer Chloe Tutton says she has had an "incredible" season after helping Britain's women's 4x100m medley relay team win European gold.