Trailing 2-0 after Elena Baltacha and Anne Keothavong lost the opening singles rubbers on Saturday, Britain decided to make a late change and put in Robson instead of Baltacha to take on Arvidsson.
Robson put in a sterling effort but was ground down 6-4 1-6 6-3 by her Swedish opponent which gave Sweden an unassailable 3-0 lead.
Robson was clearly distraught after the match and broke down in tears at the side of the court.
Her tennis in the second set was superb, but the more accurate serving of Arvidsson ultimately proved key in the decider.
Team captain Judy Murray said of Robson's performance: "We're very, very proud of her. She left nothing on the court and there was so little in it.
"We needed someone to come in and try and hit through Arvidsson and she nearly did just that. Yesterday was disappointing but today has been more encouraging after two very closely contested matches."
Keothavong lost the fourth match, a dead rubber, 7-6(6) 3-6 6-4 to Johanna Larsson.
"There is a lot to be excited about with some very positive performances form the team today and we managed to get a win in the doubles, which was a great way to finish," Murray added. "It's back now to the 15 team division, so we just need to go back there and do it all again and if we get the chance again next year we will be prepared and ready."