The team came from behind to defeat Iceland 2-1 in their playoff final in neutral Poland on Tuesday
Murray was not at his best against Paire and his faltering serve was frequently exploited by the 46th-ranked Frenchman, who caused the Briton trouble without being able to inflict any lasting damage.
He broke Murray twice in a row in the first set, which ended up going to a tiebreak that the top seed won comfortably.
Murray, who came into the championships with a niggling hip injury, still seems occasionally uncomfortable and lacking his usual zip around court.
He had too much guile for Paire, however, and wrapped up his 26th successive win against French opposition in two hours and 21 minutes.