The amount due, including damages, attorney's fees and late charges, comes out to $352,059.67, according to the court papers.
Calls for comment to Love's record company and an attorney representing American Express placed after regular business hours were not immediately returned.
The credit card company has suspended Love's charging privileges, according to the lawsuit.
Last year, Love was sued in state court for $1 million by a business management and accounting firm that claims she never paid the plaintiff its share of profits from the sale of Nirvana's publishing catalog.
In the 15 years since Cobain killed himself with a shotgun at age 27 while high on heroin, Love has remained in the spotlight with her acting and singing career and legal problems surrounding her own drug problems.
Love, who lives in Los Angeles, reportedly sold a portion of Cobain's share of Nirvana's publishing catalog for $19.5 million.