Sri Lanka, set what would have been a new Test record fourth-innings winning total of 483, were dismissed for 292 after tea on the fourth day despite fifties from Dinesh Chandimal (58), Dimuth Karunaratne (55) and captain Dhananjaya de Silva (50).
Fast bowler Atkinson, who scored his maiden first-class hundred in England's first-innings 427, led the attack with 5-62.
Player-of-the-match Atkinson secured a fifth mention on the dressing room honours boards in just his second Test at Lord's after taking 12 wickets on England debut against the West Indies in July.
He also became just the third England cricketer after Tony Greig and Ian Botham to score a century and take five wickets in an innings of the same Test.
Atkinson followed India's Vinoo Mankad (against England in 1952) and Botham (against Pakistan in 1978) as the only men to have achieved that double at Lord's.
The 26-year-old Surrey paceman's latest impressive return helped England seal a seventh successive win over Sri Lanka following their five-wicket success in last week's first Test at Old Trafford.
"To get on both honours boards is incredible," Atkinson told Sky Sports. "It will take a while to sink in."
This game was also a personal triumph for Joe Root.
For the first time in 145 matches at this level, Root made hundreds in both innings -- 143 and 103 -- to set a new record of 34 Test centuries by an England batsman.
Sri Lanka have little time to regroup before the third Test at The Oval starts on Friday.
De Silva admitted he had erred in opting to field on a good batting pitch after winning the toss.
"Definitely. I got it wrong," he told the BBC, before adding: "We have got to improve on the first innings whether we bat or bowl first, that's what I will be telling my boys is that we've got to start better."
Sri Lanka resumed Sunday on 53-2, with the odds stacked against them given the highest winning fourth-innings total in any Test is the West Indies' 418-7 against Australia at St John's in 2002/03.
But Root -- who on Saturday had taken his 200th Test catch -- reprieved Karunaratne when dropping a tough one-handed slip chance from the opener's edged cut off Atkinson.
Stone strikes
Karunaratne, 36, drove and pulled Atkinson for fours off successive deliveries on his way to a 98-ball fifty including seven boundaries.
The left-hander, however, was out shortly before lunch when injury-plagued fast bowler Olly Stone, in his first Test for three years, produced a rising 87 mph (140 kmh) delivery that Karunaratne could only glove down the legside to wicketkeeper Jamie Smith.
Chandimal counter-attacked early in the afternoon on his way to a dashing 42-ball fifty including 40 runs in boundaries before Atkinson had him caught at short leg off bat and pad.
Kamindu Mendis, boasting a colossal Test batting average of 92 following his third hundred in four Tests at Old Trafford, had made 74 in Sri Lanka's meagre first-innings 196.
But the 25-year-old left-hander, again batting low down the order, fell for just four on Sunday when caught in the slips off Atkinson.
Mendis's exit left Sri Lanka, who have now not won any of their nine Tests at Lord's, slipping towards defeat at 200-7.
De Silva kept England waiting before playing on to Atkinson, who then had Milan Rathnayake caught behind to complete his five-wicket haul.
Chris Woakes had the final say when Lahiru Kumara holed out to Stone at mid-on.
