It was Team GB's first European gold in the event since 1969 and their run of three minutes 25.05 seconds was the fastest time in the world this year.
Bundy-Davies now hopes to continue her form at the Rio Olympics in August.
"I'm hoping I can make that an Olympic medal as well at the end of the year," said the 21-year-old from Wales.
Bundy-Davies saw Great Britain home in the final leg in Amsterdam after strong running from team-mates Emily Diamond, Anyika Onuora, Eilidh Doyle.
"From the moment I got the baton I knew they had already given us a good enough lead for it to be a pretty simple job," she added.
"The girls ran absolutely amazing before me so I'm really delighted to get the European gold for the team."
Bundy-Davies has met the 400m qualifying standard for the Olympics and British Athletics are due to announce their team for Rio on Wednesday.
Legendary former hurdler Colin Jackson has earmarked Bundy-Davies as the likeliest Welsh track and field athlete to win a medal in Brazil, but the 400m runner has previously said she welcomes the "added pressure" of being tipped to succeed at Rio.
"It's kind of exciting, but scary at the same time," she said after Sunday's race.
"I'm just really looking forward to individually putting down a good performance and then hopefully winning a relay medal with these girls."
Bundy-Davies won 4x400m bronze at last year's World Championships in Beijing and claimed individual bronze at the 2015 European Indoor Championships.
Seren Bundy-Davies says she is "really delighted" after winning gold with Great Britain's women's 4x400m team at the European Championships.