The Chiefs ran in seven tries in their Pool 4 clash at the Arcul de Triumf, with Kevin Barrett, Aly Muldowney, Pat Phibbs and Ignacio Mieres all crossing, while the visitors were also awarded two penalty tries.
Fly-half Gareth Steenson settled any early jitters with a penalty, before the Chiefs won their first penalty try on 15 minutes.
The Aviva Premiership side extended their lead when Barrett crossed, while Rima Wakarua got the hosts on the board with a penalty in the last minute to reduce arrears to 17-3 at the break.
Exeter grabbed their third try shortly after the restart, wing Foster scoring, but Prato hit back through a converted try from Nicola Belardo.
However, normal service was soon resumed, a second penalty try earning the bonus point before lock Muldowney got in on the act.
Phibbs and Mieres both touched down in the final our minutes, but Exeter's win was marred by injuries to Josh Tatupu and Jack Nowell, who were both stretchered off.
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