All-rounder Stokes was ruled out for the rest of the English summer after the 33-year-old picked up a hamstring injury last week with batter Ollie Pope taking over the captaincy.
With no like-for-like replacement England have opted for Durham pace bowler Potts. The 25-year-old won the last of his six caps against Ireland at Lord's more than a year ago and has 23 wickets at an average of 29.
It does mean England have a long tail with Chris Woakes coming in at No. 7.
Crawley will also miss the series after he sustained a fractured finger and, again, England have opted for something different.
Lawrence has not played for England since Stokes and Brendon McCullum took charge, the most recent of his 11 Tests coming on the 2022 tour of the West Indies, but has often been with the squad including during last year's Ashes and the subsequent tour of India.
The 27-year-old, who averages 27 with the bat, has not previously opened in Test cricket but has done so occasionally in first-class matches and frequently in white-ball cricket, including in the Big Bash League for Melbourne Stars last season.
"One of the biggest strengths of my game is being fairly adaptable to whatever position I need to bat in," Lawrence told BBC Sport.
"I've been up and down the order for the last few years. Opening, I haven't done it for a while, but I'm really excited to get out there with Ben Duckett and hopefully put on a show."
Stokes was at England's training session on Monday and is likely to remain with the team throughout the series against Sri Lanka, but it was Pope and coach McCullum who addressed the squad.
Middle-order batter Harry Brook, 25, has been named vice-captain for the three-match series starting at Old Trafford on Wednesday.
England (v Sri Lanka): Dan Lawrence, Ben Duckett, Ollie Pope (capt), Joe Root, Harry Brook, Jamie Smith (wk), Chris Woakes, Gus Atkinson, Matthew Potts, Mark Wood, Shoaib Bashir.
