Oracle Corp.'s success in winning a $1.3 billion jury verdict against rival SAP AG sent tremors through a small but fast-growing niche in the technology sector: companies that provide technical support for other companies' software.

So-called third-party support companies supply maintenance and repairs—though not software upgrades—for a variety of business programs. The verdict Tuesday stemmed from the actions of a now-closed SAP unit called TomorrowNow that supported Oracle products. Oracle has also sued another third-party...