Sri Lanka were 81 without loss in reply before rain stopped play on day two in Galle, 411 runs behind with Nishan Madushka unbeaten on 41 and Dimuth Karunaratne 39 not out.
Lorcan Tucker scored a useful 80 while Prabath Jayasuriya, the home hero of the first Test, took five wickets for 174. Asitha Fernando and Vishwa Fernando took two wickets each.
"Scoring a Test hundred is probably something I didn't believe I could achieve," Stirling admitted.
"Scoring a hundred in the sub-continent is very special," he told reporters. "So, I am very delighted. It will mean a hell of a lot more if we can get a positive result here."
Ireland only gained Test status in 2017 and have lost all five of their matches so far.
Their previous highest score was 339 against Pakistan in their inaugural Test in 2018.
In punishingly hot and humid conditions, Stirling had retired hurt with cramps on Monday after reaching 74, but came back to the crease on Tuesday after Tucker fell.
He reached his maiden Test hundred in style, upper-cutting Asitha Fernando for six over deep point, and now has centuries in all three formats of the game.
Soon afterwards, the same bowler dismissed him for 103 as the batsman hooked a short delivery into the hands of Dhananjaya de Silva on the fine-leg boundary.
Campher was well caught by a diving De Silva at first slip to give Jayasuriya his third wicket in the innings.
Sri Lanka's openers negotiated the final session without much trouble, although Ireland's quicks found the outside edge of Madushka's bat with the new ball.
"Tomorrow is going to be a big day," said Jayasuriya. "We need to score big and take a considerable first innings lead and should be looking to bat only once.
"That should be the plan but we'll see how it goes."
Ireland lost the first Test by an innings and 280 runs.
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