The Sri Lankan skipper scored an unbeaten 61 for his 47th half-century to help his team achieve a modest 147-run target with more than 15 overs to spare for a 1-0 lead in the five-match series.
The victory was set up by Ajantha Mendis and Muttiah Muralitharan, who exposed India's batting limitations against spin.
Unorthodox spinner Mendis took 3-21 and off-spinner Muralitharan 3-37 as India were bundled out for 146 after electing to bat on a slow track.
Sri Lanka lost opener Sanath Jayasuriya (10) and Kumar Sangakkara (19) in seamer Munaf Patel's opening spell, but Jayawardene and Chamara Kapugedera (45 not out) ensured the win with a 102-run stand for the third wicket.
Jayawardene hit seven fours in his 82-ball knock and Kapugedera four boundaries in his 72-ball innings.
India were let down by their batsmen as none could get past 25 in a spineless display against a disciplined Sri Lankan pace-spin combination.
The tourists suffered a setback even before the first