The company won in the categories of best CG in film, best CG in a music video and best CG in broadcast.
The latter was reward for its work on the ITV show Primeval.
Framestore CFC has already won a Bafta and an Oscar in 2008 for its work on the film The Golden Compass.
Other Escape Awards winners included Glassworks which won for best CG in a commercial.
The Escape Awards, in partnership with Autodesk and AMD, took place at the Tate Modern in London on Thursday 27 March.
The full list of winners:
CG in Broadcast
Framestore CFC - Primeval
CG in commercials
Glassworks - Ford Flexifuel
CG in Film
Framestore CFC - The Golden Compass
CG in Music
Framestore CFC - The Salmon Dance
CG in Games
Ninja Theory - Heavenly Sword
Design Visualisation - Moving Image
Uniform - The Crystal
Design Visualisation - Still Image
Burrows Nvisage - LRX in Piccadilly Circus
Student in CG
Ian Wharton and Edward Shires - Solar