Ansys was founded in 1970 as Swanson Analysis Systems, Inc. (SASI) by John Swanson. The idea for Ansys was first conceived by Swanson while working at the Westinghouse Astronuclear Laboratory in the 1960s. At the time, engineers performed finite element analysis (FEA) by hand. Westinghouse rejected Swanson's idea to automate FEA by developing general purpose engineering software, so Swanson left the company in 1969 to develop the software on his own. He founded SASI the next year, working out of his farmhouse in Pittsburgh.

Swanson developed the initial ANSYS software on punch cards and used a mainframe computer that was rented by the hour. Westinghouse hired him as a consultant, under the condition that any code he developed for Westinghouse could also be included in the Ansys product line. Westinghouse became the first Ansys user. Swanson sold his interest in the company to venture capitalists in 1994, and the company was renamed "Ansys" after the software. Ansys went public on NASDAQ in 1996. In the 2000s, the company acquired other engineering design companies, obtaining additional technology for fluid dynamics, electronics design, and physics analysis. Ansys became a component of the NASDAQ-100 index on December 23, 2019.