Reading pushed the Swans back in the opening stages and Scott Davies flashed a long range shot just wide of the far post in the 13th minute.
Two minutes later Stephen Dobbie was pulled back after working his way into a shooting position but the visitors' wall blocked the free kick.
The Swans should have taken the lead in the 24th minute when Kalifa Cisse handled in the area. Although referee Simon Hooper waved play on, his assistant flagged for the offence but Mark Gower's weak spot kick was easily pushed away by Reading keeper Adam Federici.
With the interval approaching, the busy Dobbie tried his luck with a snap shot that had Federici diving to his right to push away for a corner.
The visitors brought on Noel Hunt at the start of the second half to boost their shot-shy forwards but neither side were able to carve out any openings and most play was confined to the centre of the field.
Reading's Hal Robson-Kanu beat two defenders before lining a shot up only to slice well wide.
The game livened up in the the 55th minute when there was a free-for-all resulting in Cisse of Reading and Ashley Williams and Angel Rangel all being booked.
This sparked the Swans into action and Dobbie went close as they began to open up the visitors defence.
They were well on top and Gorka Pintado replaced Dobbie for the final ten minutes.
Alan Tate brought a superb save from Federici following an Andrea Orlandi free kick but the flag was up for offside.
In the third minute of injury time, Morgan made a clear opening for Nathan Dyer but the winger shot wide from close range and the chance of a first league victory of the season for the Swans was lost.