England hold an unassailable 2-0 lead in the three-match Test series after wins in Galle and Kandy and Anderson will now sit out the final Test in Colombo, which gets under way on Friday.
Broad will come in as his replacement after being dropped at the start of the series.
“The thinking is that with the series sewn up it’s an opportunity to rotate,” said Anderson, who will be rested for the first time in Test cricket since 2012.
“We’re going to the West Indies (in the new year) and with a view to that, I think the feeling is they want Broady to get some Test cricket under his belt. So it’s going to be a week off (for me).
“It’s been a frustrating trip for me really because you want to contribute to the wins and I feel like I’ve tried my best but it’s not really been a series for the seamers.
“I feel like a bit of a spare part but I think that’s just the nature of playing cricket here.
“The wickets that we’ve had have obviously been very suitable for the spin bowlers, so it’s frustrating in a way for me not to have as much influence as I would have liked or have done in games elsewhere in the world, but that just the nature of playing out here.
“As frustrating as it’s been for me personally, it’s been brilliant to be a part of such a great series for the team.”
Bairstow will also return to the side in place of Curran, who suffered a side strain in Kandy and did not bowl in the second innings as England claimed a 57-run win.
Bairstow missed the first Test with an ankle injury and was replaced by Ben Foakes, who scored a century on debut in Galle and kept his place for the second Test despite Bairstow’s being available for selection.
Bairstow will return to the team as a specialist batsman and will bat at No 3 in Colombo, with Foakes keeping wicket.
England team for third Test Sri Lanka: Burns, Jennings, Bairstow, Root (c), Stokes, Buttler, Ali, Foakes (w), Rashid, Leach, Broad.