Forward Martinez scored the only goal of the game in the 31st minute against the club he nearly joined last year, acrobatically putting away Kwadwo Asamoah's cross-field ball to stun the home side at the Wanda Metropolitano.
The Argentine forward - signed by Inter from Racing Club in July - also demonstrated his blossoming partnership with Mauro Icardi early in the first half, sending his compatriot through on goal in the 11th minute.
Jan Oblak rushed from his line to smother Icardi’s chipped attempt on that occasion, but the Atletico goalkeeper could do little to prevent the game's eye-catching winner just after the half-hour mark.
Allowed to carry the ball down the left flank on the counter, Asamoah spotted Martinez free at the back post, floating over a cross that his team-mate converted in spectacular fashion.
Atletico thought they had equalised before the break when Angel Correa converted Koke’s through pass, only for the strike to be ruled out for offside by VAR. The use of technology was a relief for Inter defender Miranda, who had carelessly surrendered possession on the edge of his own penalty area to set up the opportunity for his former employers.
The home side upped the tempo after the break, Diego Costa - booked in the first half for dissent - heading over before Juanfran saw a firm drive blocked by the outstretched Dalbert.
Samir Handanovic kept out Vitolo's header with a reactionary save as Inter held on, though the result is not enough to see them overhaul Tottenham to finish the pre-season tournament in top spot.
What it means: Inter are in good shape
The 2017-18 season was one of peaks and troughs for Inter. They started strongly, suggesting a potential title challenge could be in the offing, struggled in the middle and then required a miraculous turnaround victory at Lazio to squeeze into the top four on the final weekend.
However, their additions in the transfer window have given coach Luciano Spalletti greater depth - their victory on Spanish soil was achieved without Croatian duo Sime Vrsaljko and Ivan Perisic, as well as big-name addition Radja Nainggolan.
Martinez's early promise suggests he can ease the burden on Icardi for goals, while there will be a serious competition for starting spots in the midfield.
Pat on the back: Martinez makes his mark
Atletico's loss appears set to be Inter's gain, judging by Martinez's pre-season form. The forward scored for the second game running in the International Champions Cup but demonstrated he can also be a provider too, occupying a roving role in behind focal point Icardi.
Boot up the backside: Costa worth little
It may have been a pre-season friendly, but Costa did not care. After disagreeing with a decision to award a foul against him in the first half, the Atletico striker complained so vehemently he ended up receiving a booking. It was his only real contribution as he was tightly marshalled by Inter centre-backs Miranda and Milan Skriniar during his time on the field.
Atletico face the small matter of a derby fixture against neighbours Real Madrid in midweek, as the two Spanish clubs face each other in the UEFA Super Cup in Tallinn, Estonia, on Wednesday. Inter have to wait a little longer for their next outing - they start the Serie A season with a trip to Sassuolo next Sunday.