The duo knocked out Tricia Penrose, Martin Roberts and Jenny Powell in the cookery competition and join Neil Stuke and Alex Fletcher in tonight's quarter-final.
Hamilton impressed in the final challenge with her mixed fish ragu with rice, samphire and basil, followed by an apple, ginger and Amaretti tart.
Judge Gregg Wallace commented on her dessert: "It is very good. Sugar, apple and cinnamon are the closest to a cuddle you will ever get on a plate of food."
Jackson was successful with his final dishes of chicken in a spicy coconut sauce with rice and spinach, followed by fried banana in a coconut caramel sauce with chopped fried peanuts.
Wallace praised the pudding, adding: "It tastes yummy. It tastes really yummy. Really well done. It's very well balanced."
Speaking afterwards, a tearful Hamilton said: "I'm overjoyed, of course I am. But I'm also incredibly nervous because I don't want to fluff it up in the quarter final. My mum would be so proud of me. She wouldn't have thought it possible because she hated cooking."
Jackson commented: "Getting into the quarter-final you have step it up another level. I think my fighting spirit has been ignited once again."
The first MasterChef quarter-final airs tonight at 8.30pm on BBC One.