© Daniel Hambury / PA Photos
The 23-year-old could have added a further two goals before the half-time whistle, striking the bar just two minutes after his goal, before firing wide just before the break.
The Saints looked the stronger of the two sides after the break, but had few changes to increase their lead in the opening 15 minutes of the second half.
Rodriguez crossed the ball in for Rickie Lambert in the 60th minute, but the striker could not stretch far enough to connect with the pass.
Adam Lallana doubled the visitors' advantage, calmly knocking the ball past Federici in the 72nd minute, after Rodriguez picked out the substitute with a high pass.
Reading thought they had got one back four minutes later, as Boruc caught a Le Fondre strike, but his momentum carried him backwards into the net, although replays did not show whether all of the ball crossed the line.
Mauricio Pochettino's side had several chances to score a third in the final minutes of the match, but they were unable to find the target again.
The result moves Southampton up to 11th in the Premier League table, while Reading remain bottom.