Martinez's stunner in the 31st minute proved enough for Inter to triumph on Saturday night as both teams brought their preparations for the 2018-19 campaign to a close.
Atletico will now march on to Wednesday's European Super Cup with Real Madrid, while Inter will open their 2018-19 Serie A campaign away to Sassuolo next weekend.
Inter had the first serious chance of the match in the 11th minute, but Mauro Icardi - having skipped past the challenge of Diego Godin inside the box - could not find a route past Atletico goalkeeper Jan Oblak.
Marcelo Brozovic then stung the palms of Oblak with a long-range strike three minutes later as Inter looked the team more likely to make the breakthrough in the Spanish capital.
Inter's next chance arrived in the 20th minute when Matteo Politano struck one from distance, and the 25-year-old's effort only just missed the Atletico goal.
It might have been a pre-season friendly, but there was a real competitiveness to the match, and Diego Costa's frustrations boiled over in the 24th minute, which led to the striker being booked following a word with the referee.
Inter made the breakthrough in the 31st minute, and it was a stunning volley from summer arrival Martinez, who acrobatically found the back of the net with a wonderful effort at the far post.
Godin headed over the Inter crossbar in the period that followed as Atletico looked to level, before Filipe Luis struck into the stands.
Atletico looked to have equalised in the final stages of the first period when Angel Correa converted from inside the Inter box, but the goal was disallowed for offside following a VAR review, with the correct decision reached.
Costa headed over the Inter crossbar early in the second period as Atletico started on the front foot, while Juanfran had one deflected wide of the away side's post with Inter coming under severe pressure.
The visitors remained a threat, however, and Martinez headed just wide of the Atletico goal just before the hour mark.
The tempo suffered due to the sheer number of substitutions, but Atletico were the team on the front foot as the pre-season friendly headed for its final 10 minutes of action.
ATLETICO MADRID (4-4-2): Oblak; Juanfran (Arias 61'), Godin (Montero 62'), Gimenez (Savic 46'), Filipe (Lucas 62'); Thomas (Riquelme 46'), Koke (Martins 62'), Saul (Vitolo 62'), Lemar (Olabe 62'); Correa (Griezmann 61'), Costa (Kalinic 62')
INTER MILAN (4-2-3-1): Handanovic; D'Ambrosio (Candreva 81'), Miranda (De Vrij 60'), Skriniar, Dalbert; Vecino (Candreva 81'), Asamoah (Gagliardini 60'); Politano (Mario 71'), Brozovic (Valero 60'), Lautaro (Karamoh 71'); Icardi