The animation looked like an Anime. It was about this kid that is playing this video game, where some dinosaur monsters are chasing a girl. If I'm not mistaken, he's playing as the monsters, but then he gets sucked into the video game and realizes the girl is on the side of good or something. He tries protecting her, and then he makes friends with a monster pet of some sort, whom he names after a dish his mom makes (I forgot the name of the dish/monster) because the monster looks very similar (it's white, with a green leaf/something on top of it).
As a specific, the boy is NOT permanently trapped in the game. He can get out of the game periodically, but every time he plays it, he travels to the dimension (and a stock footage of him getting sucked in plays).
I remember watching two or three episodes as a kid, but then they stopped airing it. It's probably from the 90s or early 2000s at most. I'm pretty sure it was an anime, but then again, Martin Mystery had "Animesque" animation too, so maybe it's American? I dunno.