Halfway through the morning foursomes session, it looked like the points would be split, with Big 10 individual champion David Nyfjall and Hugo Townsend up in their match, and Luis Masaveu and Quim Vidal winning theirs.
A birdie from Townsend and Nyfjall halted the Spanish comeback and took the score to 1-1 at lunch.
Spain went up early in four of the afternoon singles and never looked in trouble.
Javier Barcos won his match 4&2, shortly followed by Luis Masaveu winning 3&2. David Lundgren responded with a point for the Swedish side, but Ballester was two-up in his match, while Puig was four-up in his.
A birdie on the par-3 16th from Ballester secured a 3&2 victory, and the title for the Spanish side.
The victory is their fifth in the European Amateur Team Championship and their first since 2017.
In the bronze medal match, defending champions, Denmark took on hosts, England.
Rasmus Neergaard-Petersen and Søren Lind won the first foursomes match 3&1 to draw first blood for Denmark. The second foursomes match had to be decided in extra holes. The point would ultimately go to Denmark on the 20th, with Hamish Brown and Jens Kristian Thysted making it 2-0.
The hosts struggled to recover from the two-point deficit to the defending champions. Frederik Birkelund defeated Arron Edwards-Hill 3&2 to put Denmark within just a point of securing third place. Despite some close matches in the afternoon, Frederik Kjettrup defeated Barclay Brown on the 19th to clinch the bronze medals for Denmark.