Mahela Jayawardene's 19th one-day century rescued the 1996 champions from a perilous 51 for four in pursuit of a 233-run target after openers Lahiru Thirimanne and Tillakaratne Dilshan both fell for golden ducks.
Jayawardene (100) and Mathews (44) put on a crucial 126 for the fifth wicket to steady the ship before Thisara Perera smashed a quickfire 47 not out to see them home.
It was 1996 champions Sri Lanka's first win of this tournament after an opening 98-run loss to New Zealand, while debutants Afghanistan have now been defeated twice following a first-up 105-run drubbing by Bangladesh.
"I am feeling relieved at the moment but I thought we came back well. It was a see-saw kind of game. They started off well and then we pulled it back in the end. I knew I had to stay out there with Mahela to get some kind of stand going."
Afghanistan captain Mohammad Nabi admitted his team's 232 all out was always going to be hard to defend.
Jayawardene, who was named man-of-the-match, praised Afghanistan and said the gap between Test nations and the second tier Associate sides was narrowing all the time.
"All credit to Afghanistan. They batted really well in tough conditions, then followed it up beautifully with the ball," said the 37-year-old.
"They have pace, movement and they're a handful. There is not a big gap between the top countries and these guys.
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