The latter, still returning to full fitness after his hamstring injury last season, had the first effort of the game in the 15th minute, having a close-range shot blocked by a defender after Martin Cranie won a corner at the back post.
Kachunga then combined well with Depoitre for an acrobatic volley five minutes later – only to see his shot well blocked by a Girona defender.
With Town dominating, it was only a matter of time before the deadlock was broken and came via a smart counter-attack for Wagner's men.
Stand-in left back Regan Booty released Kachunga down the left channel who drove at the defence before setting up Palmer to strike home sweetly from 15 yards out.
Town continued to impress in the second-half - Depoitre firing over after Dean Whithead forced the opposition into an error inside their own half but couldn't find a second goal.
At the other end, goalkeeper Joel Coleman commanded his area well and dealt with the Girona threat as they searched for an equaliser late on.
Huddersfield Town Team (4-2-3-1): Joel Coleman; Regan Booty, Martin Cranie, Mark Hudson, Harry Bunn; Danny Williams, Dean Whitehead (c); Rajiv van La Parra, Kasey Palmer (Jack Payne, 60), Elias Kachunga; Laurent Depoitre
Subs Not Used: Ryan Schofield, Rarmani Edmonds-Green, Danny Kane, Jordan Williams, Lewis O’Brien, Sean Scannell.