Haaland, who has scored his side's opener for the third season in a row, scored from a penalty in the first half and then a made a well-timed run in the second half for his second goal.
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"It was good. [It was a] good start, there is nothing much to say other than that, and it was 2-0. Good to get some minutes in the legs early in the season and we have to keep going," Haaland said after the game.
"As expected, we saw the potential against Bayern [Munich in preseason] and it was a bit down against Liverpool. It will come with games; it should have been more today."
He gave away the penalty just six minutes later. Haaland arrived in the box and the goalkeeper mistimed his challenge for the ball, taking out the 22-year-old.
Haaland stepped up to take the penalty and fired it into the far left of the net, sending Areola the wrong way.
His second goal came from a sublime pass from midfield from Kevin De Bruyne, who put the ball in behind the West Ham defence. Haaland ran onto it and slid it across Areola and into the far corner.
"Yes. I should have been there [to connect with Ilkay Gundogan's cross]. It's about the connections we do every day in training and practicing, so we get better and this will come," Haaland added when asked if he should have scored more than two goals.
"There were some good celebrations, so I'm happy. Nice that he [his father, Alfie] saw both goals as it is a big moment for me as a debutant in this competition.
"Also now it is almost 30 minutes since I scored the last goal, so I have to keep going."