Barrett headed a Simon Francis free-kick into the top left-hand corner from eight yards in the 54th minute and midfielder Christophe blasted home a 20-yard shot into the opposite corner in the 89th minute after being teed up by Franck Moussa.
Cheltenham started brightly and should have opened the scoring in the ninth minute when Yuri Berchiche crossed from the left byline, but striker Michael Antonio hooked the ball over the bar from six yards out.
Debutant Robins goalkeeper Will Puddy then produced some decent saves to deny Southend forward Theo Robinson, who also had a goal ruled out for a handball.
Cheltenham nearly found a quick equaliser four minutes after Barrett's goal, but Blues keeper Steve Mildenhall dived to save David Hutton's free-kick.
And the Shrimpers came within a whisker of doubling their advantage when Lee Barnard picked up a poor Puddy clearance and tried an audacious 40-yard shot which went just wide before Christophe's clincher.