Gadir struck in the 54th minute for the only goal of the game as Maccabi put aside their poor recent form with a morale-boosting victory.
Larnaca's hopes of getting anything out of the match all but ended when they were reduced to 10 men with nine minutes remaining when Ander Murillo was shown a straight red card.
The result leaves Maccabi top of the opening-day standings in Group J with Steaua Bucharest and Schalke drawing 0-0 in Thursday's other game in the pool.
Elisha Levi's Maccabi side have had a disappointing time of it recently, being knocked out of the Champions League play-offs by Genk on penalties and since then failing to win either of their domestic games.
Saturday's embarrassing 1-0 defeat at lowly Ashdod sent Maccabi reeling to eighth place in the standings, but they received a boost with a largely dominant performance against Larnaca.
Maccabi were soon creating chances against their Cypriot visitors, with Weam Amasha testing Marco Fortin in the eighth minute before the visiting goalkeeper was called on to keep out efforts from Idan Vered and Yaniv Katan.
Maccabi were looking keen to shoot whenever they got the chance and in a seven-minute spell around the half-hour mark Gadir, Amasha (twice), Vered and Eyal Golsa all tried their luck from range, although none of them managed to hit the target.
At the other end, home goalkeeper Bojan Saranov had little to be concerned about aside from a couple of early tries from Kyriakos Pavlou and a Murillo shot just before half-time that went wide.
Miljan Mrdakovic also fired wide for the visitors at the start of the second half before Maccabi took the lead through Gadir, who headed in Taleb Tawatcha's cross for the match-winning goal.
The hosts came close to extending their lead in the 67th minute but Fortin was able to parry Katan's shot.
Larnaca were struggling to find a way through the home defence as they sought an equaliser and, after Fortin had denied Marlon De Jesus, the visitors suffered a further blow when Spaniard Murillo was dismissed for a professional foul on De Jesus.
There was no way back for Larnaca after that, although they did manage one late effort with Gonzalo Garcia firing wide in stoppage time.