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They added 96 for the third wicket at Chester-le-Street after Alastair Cook (15) fell five short of reaching 10,000 Test runs and under-pressure Nick Compton (9) continued his poor run.
Hales, seeking his first Test hundred, was caught at slip, while Root spooned to cover as Sri Lanka fought back.
But Jonny Bairstow and Moeen Ali added 70 to give England the upper hand.
Bairstow was caught behind for 48 to give the excellent Nuwan Pradeep his third wicket, but Moeen remains unbeaten on 28.
The hosts won the first Test at Headingley by an innings and 88 runs.
Despite some superb catching from Sri Lanka on a cold day, England will be confident of amassing the sort of first-innings total that will put pressure on the tourists' inexperienced batting line-up.
Hales made 86 at Headingley and picked up where he left off with another sizeable contribution at the top of the order.
After a circumspect start, he accelerated after lunch in partnership with Root, scoring heavily down the ground on the off side.
He looked poised to become the first Englishman to make an international century in all three forms of the game before he swung hard at the part-time left-arm spin of Milinda Siriwardana and was brilliantly caught at slip by Sri Lanka captain Angelo Mathews.
Nonetheless, Hales is beginning to look like he could be the reliable opening partner for Cook England have been looking for since Andrew Strauss retired in 2012.
For England's other top-order batsmen, it was a tale of familiar flaws as once again the team made a stuttering start to their innings after Cook won the toss.
The skipper was first to fall, in characteristic fashion, fiddling needlessly outside off stump and offering a sharp catch to Dimuth Karunaratne at second slip.
Compton failed to dispel the question marks over his place during another scratchy and underwhelming innings, grinding his way to nine off 34 balls before before top-edging to long leg.
Even Root could reproach himself for a poor shot, offering a simple catch to cover as he attempted to work a short ball from Pradeep to leg.
It was the seventh time in the past eight innings in which Root has passed 50 that he has failed to go on to three figures.
Although England ended the day narrowly on top, this was a fine performance in the field by Sri Lanka.
Their catching was magnificent - as well as Mathews and Karunaratne, Suranga Lakmal produced a brilliant tumbling effort at long leg to dismiss Compton and Lahiru Thirimanne flung himself to his right at cover to end James Vince's innings.
They also performed strongly with the ball, despite missing two of their best pacemen, Dushmantha Chameera and Dhammika Prasad, through injury.
Pradeep and Lakmal bowled a disciplined line and length, Siriwardana chipped in with two wickets, and they conceded only four extras in 90 overs.
England opener Alex Hales on BBC Test Match Special: "I'm pleased with the form I'm in. When you get in you want to make a big score, but I'd have taken it at the start of the day.
"It's was my own fault - I probably try to hit it too hard. Mathews went a little deeper and took a brilliant catch.
"This is a better pitch to bat on than Headingley so I felt a little bit more fluent today. There's still things to work on and I feel like I'm improving every day.
"This has been a learning curve. I'm enjoying being as good as I can in all three formats."
Sri Lanka all-rounder Milinda Siriwardana: "It was a brilliant day for us. It's not easy to get wickets, so we're happy. We fielded really well. We took the half-chances.
"The conditions are better than the first game. It will turn and it's good for the batsmen.
"It was cold this morning. I had the hand-warmers in my pocket. It was the coldest I've played cricket in."
Listen to Jonathan Agnew and Geoffrey Boycott review the day's play on the TMS podcast.
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Alex Hales hit 83 and Joe Root 80 as England reached 310-6 against Sri Lanka on day one of the second Test.