Both teams had already sealed a place in the quarter-finals but it was the French side which prevailed at Kingsholm, finishing the pool stage with a perfect record of six wins from six.
Jeremy Thrush's try opened the scoring for Gloucester, but two Colin Slade penalties and a Steffon Armitage try gave Pau a 13-7 half-time advantage.
Billy Twelvetrees cut the gap to three points five minutes into the second half, however, any hope of a turnaround was extinguished as Pau ran in three more tries.
Frank Halai, Lourens Adriaanse and Daniel Ramsay all crossed for Pau, putting the game beyond doubt despite Callum Braley's try for the host.
Jake Polledri went over late on for Gloucester but his score served as nothing more than a consolation.