Minutes later Ki Sung-Yueng tried his luck with a free kick from distance only to see it sail high over the Reading crossbar.
Swansea settled into their passing rhythm and kept the ball out of the reach of Reading, whose first real attack of the match came in the 16th minute.
Michu had another good chance to put the Swans in the lead just before the half-hour mark when Chris Gunter inadvertently chested the ball into his path, but the Spanish striker shot high from inside the six-yard box.
The game was more even in the opening minutes of the second half, as the Swans failed to keep up their attacking game in the first 45 minutes.
Reading thought that they had broken the deadlock in the 73rd minute when Adam Le Fondre turned in Harte's delivery, but he was adjudged to have punched the ball in.
The Royals went close again with six minutes left on the clock, but Chico kept out Le Fondre's volley with a headed clearance off the line.