Surrey wicketkeeper Smith's 111 was the centrepiece of England's 358 all out -- a total that left them 122 runs ahead of Sri Lanka's 236 on the third day of this three-Test series.
Tailender Wood made a typically quickfire 22 off 13 balls, including three fours and a superb hooked six over square leg off Asitha Fernando that was caught one-handed by a spectator in the crowd holding a pint in his other hand.
Asitha Fernando, however, had his revenge when he bowled Wood as he backed away and the paceman finished the innings with figures of 4-103 in 18 overs.
England resumed on 259-6, with Smith -- playing just his fourth Test -- 72 not out overnight after falling agonisingly short of a century last time out on England duty while making 95 against the West Indies at Edgbaston in July.
Together with Harry Brook (56) and Chris Woakes (25), Smith had shared stands of 62 and 52 respectively on Thursday before both batsmen were bowled by superb, sharply turning deliveries from left-arm spinner Prabath Jayasuriya.
Once again Smith demonstrated his ability to both bat with an established batsman and then maintain a good tempo with the lower order and he enjoyed another useful stand of 66 with Surrey team-mate Gus Atkinson on Friday.
A slow outfield, allied to some disciplined bowling, meant an England team known for their aggressive batting had to fight hard for their runs on Thursday.
But the strong crosswind once more blowing across Old Trafford made conditions awkward for Sri Lanka's bowlers as well.
Asitha Fernando had done the bulk of the damage as England faltered at 67-3 on Thursday.
Smith, however, drove his first ball Friday down the ground for a boundary straight out of the coaching manual to go to 80 and five deliveries later he struck him for an equally textbook four through extra cover.
And he completed a 136-ball century, including seven fours and a six, by confidently clipping paceman Milan Rathnayake for two off his pads.
Sri Lanka eventually broke the seventh-wicket stand when Atkinson (20) glanced Rathnayake down the legside to wicketkeeper Dinesh Chandimal.
And soon afterwards England were 315-8 when Smith toe-ended an intended cut off Jayasuriya, with Chandimal holding a tough catch to dismiss his opposing gloveman.
Jayasuriya took an economical 3-85 in 31 overs, with Matthew Potts last man out when caught in the deep off Vishwa Fernando.
