Kendall Waston was sent off after the referee used the VAR system on 13 minutes, and Martínez stepped up and slotted home the penalty to break the deadlock.
Aaron Maund turned into his own net just before the hour mark before Martínez scored his second of the game, just two minutes later.
Erik Hurtado pulled a goal back for Vancouver five minutes from time, but Martínez's header, three minutes later, wrapped up the three points for Atlanta.
After a disappointing opening weekend to the new season, Atlanta bounced back and beat D.C. United 3-1 in their first match at their new stadium.
Atlanta break deadlock early
The Five Stripes had the first chance of the game on eight minutes when a free-kick was delivered into the box from the left side.
The ball ricocheted of Efraín Juárez and then Waston before dropping to the feet of the unmarked Franco Escobar, who was denied by a fantastic reaction save.
However, Leandro González Pirez was lying on the floor and after the referee used VAR, he found that Waston had fouled the Atlanta man and the hosts were awarded a penalty five minutes later.
After the lengthy delay, Martínez stepped up and slotted home the spot-kick into the bottom corner to open the scoring.
The home side should've doubled their lead two minutes from half-time. Juárez was dispossessed in Atlanta territory and Julian Gressel then played the ball through to Héctor Villalba on the right side.
Villalba drove forward and inside before crossing low across the penalty area for Miguel Almirón, who somehow managed to strike wide at the back post from just a few yards out.
Two quick goals put hosts in complete control
Just before the hour mark, Atlanta managed to double their advantage in the game after some free-flowing football in and around the edge of the area.
Villalba threaded the ball to the right side of the box for Gressel, who drilled the ball low and unfortunately for Vancouver, Maund turned the ball into his own net.
The Five Stripes managed to catch their opponents out against, just two minutes later as Almirón played the ball through to the advancing Martínez.
The Venezuelan striker burst through on goal before producing a composed finish, slotting home into the bottom corner.
Vancouver comes within inches of pulling a goal back on 72 minutes, with Brek Shea moving the ball to Erik Hurtado, who then found in the area, Yordy Reyna.
Reyna gathered the ball and then struck low across goal and off the foot of the post before Atlanta scrambled the ball clear.
Martínez wraps up hat-trick minutes after Vancouver consolation
The Whitecaps did manage to get on the scoresheet on 85 minutes though, with Hurtado latching onto a long ball over the top before lofting the ball past the approaching 'keeper and into the net.
Vancouver's hopes of a comeback were ended three minutes later as Andrew Carleton's delightful lofted cross was steered home by Martínez at the back post.