After an entertaining opening 45 minutes it was the Royals who opened the scoring when Nick Blackman’s thunderous 25-yard drive cannoned into the back of the net off the underside of the crossbar less than a minute after the interval.
Bartley restored parity within two minutes, converting Jefferson Montero’s driven cross in a frenetic start to the second half.
Swansea finished the game the stronger of the two sides and were denied victory when both Jonjo Shelvey and Marvin Emnes hit the woodwork.
However, neither side could find the breakthrough on a rain-soaked evening in Buckinghamshire and the spoils were shared.
Garry Monk’s side kick off their league campaign away to defending champions Chelsea on August 8, while Reading start their Sky Bet Championship campaign away to Birmingham City.