Ward opened his campaign with a 6-1 1-6 6-0 victory over Irishman James McGee on Tuesday but failed to make it any further on the clay of Roland Garros after Lindell beat him 3-6 7-6 (7-5) 6-2.
The 28-year-old deservedly took the opening set but narrowly lost out in the second on a tie-break as his opponent, ranked 94 places lower at 200, went on to cruise through the decider with two breaks in the final four games on a rain-interrupted day in Paris.
Meanwhile, Liam Broady’s dreams were ended after an agonising defeat to Radu Albot in the first round.
Leading 5-4 overnight, Broady took the first set on a tie-break but then missed two match points at 6-5 in second as Albot battled back to win 6-7 (5-7) 7-6 (7-2) 6-2.
Great Britain still have Brydan Klein and Kyle Edmund, who are in second-round action on Thursday, bidding to join Andy Murray and Slovenian-born Brit Aljaz Bedene in the main draw.
There was some good news in the women’s draw as Johanna Konta and Naomi Broady both won their first-round matches.
British No 2 Konta beat Jovana Jaksic from Serbia 6-0 6-3 while Broady defeated Turkey’s Cagla Buyukakcay 6-1 7-6 (7-4).
World No 43 Heather Watson qualified directly to the main draw which takes place on Friday.