Yes, a touchdown was scored.
Work Step by Step
The Saints were 22 yards from the goal. They gained 5 yards with the first down, putting them 17 yards away. They then lost 10 yards with the second down, putting them 27 yards away. They lost another 2 yards with the third down, putting them 29 yards from the goal. They gained those 29 yards with the fourth down, scoring a touchdown.