Disclaimer: I have not watched the movie, I have only read the books.
Haymitch didn't really convince the Gamemakers to allow two tributes to win, and the input he had over the packages was primarily when they would be delivered, and what they were.
Packages being delivered to tributes was a regular part of the Hunger Games. The whole point of making a big presentation of the tributes, from the costumes to the ratings assigned by the Gamemakers, was to establish just how popular each individual tribute was with the crowd. The more popular a tribute was, the more likely they were to receive packages, and the more expensive those packages could be. Cinna probably deserves more credit for getting Katniss her packages than Haymitch, since Cinna's costumes were such a smash hit. However, as their mentor, Haymitch gets to decide when the packages are delivered (and probably has some undetermined influence over what form the packages take).
As for the decision to allow two tributes to win, Katniss gets most of the credit. She was the one who set up the possibility of all tributes dying, a situation which is problematic for the Gamemakers, particularly when the popularity for Katniss and Peeta was so unusually high (due to the "tragic lovers" story they had been playing up). Allowing both of the remaining tributes to commit suicide would have also been a political blow to the government, as it would mean that Katniss and Peeta "beat" the Capital at its own game. It was decided that it was preferable for the Capital to be seen as generous in bending the rules to allow both of them to survive, and it would be for the immensely popular tributes to succeed in openly defying the rules of the game.
That's not to say Haymitch had no role in this. In the later books it is revealed that Haymitch won his Hunger Game by virtue of being very intelligent. He was a careful planner, even after he descended into alcoholism, and was no doubt making key comments at precisely the right time to help nudge the Gamemakers and Capital representatives to make the decisions that would be most beneficial for the District 12 tributes.