Greaves was the last man out on Monday (Tuesday AEST) as West Indies made 499 in reply to Sri Lanka's 9-549.
The tourists lost first-innings century-maker Lahiru Udara to a four-ball duck and Nishan Fernando for 20 before reaching 2-92 at stumps, increasing their overall lead to 142 with a day remaining.
Dinesh Chandimal was 40 and Kamindu Mendis 30 not out in a partnership worth 60 runs by close of play at Sir Vivian Richards Stadium in Antigua and Barbuda.
West Indies won the first of the two-Test series by an innings and 217 runs and need only a draw to claim a first series win in the World Test Championship since beating Bangladesh in the Caribbean in 2022. On a pitch on which only 21 wickets have fallen in four days and 1140 runs have been scored, the odds are in favour of the home team.
Greaves achieved his third Test century and second highest score in a partnership of 242 runs for West Indies' fifth wicket with Shai Hope, who made 112. The stand was reminiscent of the 196 the pair put on against New Zealand in December when Hope made 140 and Greaves 202 not out.
Hope and Greaves' latest partnership guided West Indies from 4-144 to 5-386.
Sri Lanka's bowlers kept up pressure throughout the day, making good use of short-pitched bowling to limit the scoring. Asitha Fernando took 5-130 from 40 overs, his third five-wicket bag in Tests, and Prabath Jayasuriya 3-131.
The West Indies resumed on 4-318 and Hope should have been out without adding to his overnight score of 86 when he edged Milan Rathnayake's second ball down the leg side. However, Sri Lanka didn't review the on-field umpire's not out decision.
Hope missed the first Test due to a shoulder injury in training, but returned and knocked off his third century in his last five Tests off 199 balls before his was out stumped down leg trying to pad away left-arm spinner Sonal Dinusha.
Greaves resumed on 85 and raised his second Test hundred in Antigua off 201 balls with a quick single.
