At tea on a rain-marred fourth day, England, 2-0 up in the three-match series, were 206/6 in their second innings.
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Rain meant only 12.2 overs were possible on the final day, with Sri Lanka 78 for 1 in their chase of 362 when play was finally abandoned.
Bayliss praised Hales, whose position was under scrutiny after he struggled in the tour of South Africa, but admitted the badly out-of-form Nick Compton faces an uncertain future in the Test line-up.
Bairstow repeatedly helped England recover from top-order collapses against Sri Lanka and Cook likened his contribution to that of retired former England wicket-keeper Matt Prior.
The Sri Lanka series also saw wicket-keeper Bairstow hold 19 catches, although doubts about his glovework persisted after he grassed a couple of routine chances.
“Alex Hales has tightened up his game from South Africa and learned about Test cricket”, Cook said.
As Sri Lanka players surveyed the scene, there was no sign of the national flag that had been draped across their dressing room balcony on Sunday in protest at a wrongly called no-ball that denied them a key wicket.
Compton had been promoted to open in place of England captain Alastair Cook, after the left-hander suffered a knee injury fielding close in earlier on Saturday.
But it was not to be, England having to resort to spin at each end in classic seam-bowling conditions as worsening light compromised them even before rain returned for the last time.
But Anderson, England’s all-time leading Test-wicket taker, struck when Silva, padding up after failing to pick an inswinger going against the Lord’s slipe, was lbw for 16. Moin Ali was the bowler and Cook made a decision to stand close to the batsman to put more pressure.
The accolades were all his after the wicketkeeper-batsman consolidated the hundred he completed the previous evening by finishing unbeaten on a Test-best unbeaten 167 out of 416 all out.
But by then he had already survived one controversial let-off when he was bowled by Nuwan Pradeep on 58.
Bairstow was soon the one who could only hope his mistake did not prove so costly, after letting Woakes down when he dropped Karunaratne off the first ball of the all-rounder’s spell.
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Cook must instead wait overnight to discover whether he is fit to bat again here – and at least on an eventful third day of this third Test, England recovered some composure in Alex Hales and Jonny Bairstow’s half-century stand before closing on 109 for four and a lead of 237.
