Cook ended a 12-year vigil at the top of the order with a farewell century against India earlier this month, leaving a yawning hole in the side.
Surrey's title-winning captain Burns was overlooked over the course of five prolific years by Smith's predecessors but the 28-year-old will now get the chance to succeed the country's record run-scorer.
He is one of three opening batsmen selected for the three-match series, alongside fellow Test newcomer Joe Denly and Keaton Jennings.
The latter, despite being the man in possession and Cook's final partner, probably had the least compelling case to travel following a confidence-sapping summer.
Speaking after a Trent Bridge meeting with fellow selector James Taylor, captain Joe Root and coach Trevor Bayliss, who chipped in via Skype, Smith said: "I don't really get hung up on the 'next Alastair Cook' narrative. I don't think that's the way you select.
"The fact is there is a vacancy at the top of the order and there will be a slightly new look, inevitably, because of Alastair going.
"If (the best player) was right-handed and slogged sixes all the time and that was the best thing to do, we'd go with that.
"But I think Rory Burns really deserves his chance in the Test squad. He has impressed everyone with the consistency of his run scoring, not just this year when he's scored 1,300 runs and led Surrey to the title – which are both significant achievements – but over five years, when he has consistently scored over a thousand runs (per season)."
Denly's road has been different. He played nine one-day internationals and five Twenty20s in 2009-10 but has spent the intervening eight years firmly off the radar.
He rebuilt his confidence as a roving hand for hire on the global T20 circuit and has done enough at Kent – batting at three and also impressing with the kind of part-time leg-spin which could be useful on the sub-continent – to earn a second life at the age of 32.
Smith said: "I played with Joe when he was a very young man, you could see he was a fantastic natural athlete, a brilliant mover, talented batter, bowled leg-spin and everything came very easily to him.
"I think with players who get opportunities late in their careers you just want to know that touch of class is there, and there definitely is with him."
Smith stood firm in support of Jennings, who averaged 19.20 in six Tests this summer and may owe his selection to a debut century in Mumbai in similar conditions to those England will face in Sri Lanka.
"Like every opening batsman this summer Keaton found opening the batting difficult, there's no secret about that, but when it comes to selecting a touring party we felt he was in our best squad to beat Sri Lanka," he said.
"We are here talking about how are you going to replace Alastair Cook and what a hole he leaves – it's a wonderful career and deserves every superlative going – but he didn't find opening the batting in England in 2018 very easy.
"(Jennings) has come through a really tough summer and sometimes in international sport getting through something difficult can put you in good stead."
Warwickshire's Olly Stone continued his sharp career trajectory, following his first one-day call-up for next month's five-match series with a maiden Test spot.
The 24-year-old's ability to breach 90mph and an impressive statistical haul in Division Two this year counted in his favour but he appears alongside senior seamers James Anderson and Stuart Broad – not instead.
On rumours one or both might have been spared the trip, Smith added: "At some stage there will be some need for succession planning, however in this instance we felt it was perfectly manageable for them to play these three Tests."