Platt, a pounds 4.75m buy from Sampdoria, played just the first half at Roots Hall but took only seven minutes to score, with an impressive half-volley from 18 yards after Bergkamp had cut back a cross from the left.
Platt was protecting a hamstring injury which restricted him to only one appearance in Arsenal's three-game tour of Sweden last week.
Ray Parlour added Arsenal's second goal 11 minutes before half-time and the spirits of the north London club's 4,000 travelling fans in a crowd of 8,923 were lifted again in the 72nd minute when Bergkamp set up fellow Dutchman and second-half substitute Glenn Helder for the fiercely-struck third.
One blemish on the Arsenal performance was the booking of Ian Wright, who collected his second yellow card in four pre-season games after a flare-up with the Southend full-back Mark Hone.
The Premiership side's defence, comfortably in control for most of the game, lost concentration late in the game when the Southend substitute Dominic Iorfa scored 13 minutes from the end.