Dilruwan Perera took four wickets and Malinda Pushpakumara three wickets to see off England in the final session of day one after they elected to bat first in Kandy.
After Adil Rashid struck twice to England's tourists' hopes of securing a lead on the turning pitch, Silva built crucial partnerships including a 56-run ninth wicket stand with Akila Dananjaya as the hosts edged past England's total.
Leach then made the breakthrough at the other end, getting a ball to break sharply off the surface and beat Kaushal Silva's forward defence, to rattle off stump - the batsman dismissed for 6.
Root was punished after he "appeared to show dissent" by shaking his head and kicking the ground in the 76th over of Sri Lanka's innings after umpire Marais Erasmus adjudged Dilruwan Perera was not out to a delivery from Moeen Ali.
"Dimuth (Karunaratne) was superb today with the bat and it was extremely unfortunate that he had to fall for a brilliant catch".
"We're very much feeling good about the fact that we're bowling last on that wicket. But we still have got a batter at the crease", a confident Mathews told reporters after an intriguing day's play.
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Not surprisingly, all Sri Lankan wickets went to the spinners.
Jennings' combo catch with Foakes was his second spectacular piece of fielding at short leg on the day, the Lancashire batsman having already taken a stunning one-handed grab to dismiss Dhananjaya de Silva.
"We're a little bit disappointed that they got a 40-run lead", Leach told Sky Sports.
According to global cricket rules an umpire can dock five runs if he feels the batsman ran a short run deliberately.
Karunaratne perished for 63 scored off 125 balls (4 fours) and nine runs later England had another wicket when Stokes was again in the act plucking a blinder of a catch at slip to send Kusal Mendis packing off for one.
"Those things can happen its part of the game we don't blame him".
Bowling: Anderson 14-2-40-0, Curran 4-0-19-0, Leach 29-5-70-3, Ali 25-1-85-2, Rashid 22-2-76-3, Root 8-0-26-1, Stokes 1-0-9-0.