Ward, 27, took the first set 6-4 on his fourth set point on the Robredo serve, before the players were forced to leave the court following a downpour at Roland Garros.
Ward booked his spot in the first round by overcoming Blaz Rola of Slovenia in qualifying, but Robredo ultimately proved too strong on clay this afternoon.
The Spaniard broke in the opening game of the second set and then once more at 4-4 before serving for the second set after 43 minutes of tennis.
Robredo then took the third set 6-2 to move ahead in the match, before securing a single break in the fourth on his way to booking a spot in round two.
Robredo made the last eight in Paris last year.