Opening batsman Gambhir (62) and Raina (54) added 113 runs for the second wicket after India had lost Sachin Tendulkar (five) cheaply to a dubious leg-before decision while chasing 247 for victory.
Dhoni (61*) then took command, putting on 44 runs for the fourth wicket with Yuvraj Singh (23) and 66 runs with Rohit Sharma (25*) as India overhauled the target with just under two overs to spare.
Sanath Jayasuriya (107) had compiled his 28th ODI century and Kumar Sangakkara made 44 in difficult conditions as Sri Lanka finished with a decent 246 for seven after being asked to bat first.
Sri Lanka then claimed the early breakthrough when left-arm seamer Thilan Thushara rapped Tendulkar on the pads with a delivery that pitched just outside leg stump and straightened.
Kumar Dharmasena, umpiring his first ODI, upheld a vociferous appeal, although replays showed the height was questionable.
Left-hand batsman Gambhir, opening in place of Virender Sehwag who was ruled out due to injury, then steadied the innings in the company of Raina.
An accurate line from Thushara and Nuwan Kulasekara had kept the batsman quiet, but Raina broke the shackles by hitting Kulasekara for a six and two boundaries in the bowler´s fourth over.
The two easily tackled Sri Lanka´s feared spinners Muttiah Muralitharan and Ajantha Mendis, and went on to amass half-centuries.
Gambhir then threw it all away as he attempted to hit Murali over the top only to be caught at long-off, while Raina was run out shortly after while attempting a second run off Mendis.
Dhoni promoted himself up the order and turned in a masterful performance.
He built useful partnerships with Yuvraj and Rohit to guide the team home.
Earlier, Dhoni´s decision to field at first seemed justified when Zaheer Khan, handling the new ball, began with two maidens and opener Tillekaratne Dilshan (zero) was sent back early.
Dilshan was the victim of an unfortunate mix-up and was run out for a duck after facing just three deliveries from Zaheer.
The Sri Lankan opening batsman had shaped to defend a delivery from Zaheer, but an inside edge on to his pads had looped to square-leg.
Dilshan set off for a single, but was sent back by Jayasuriya when more than half-way down the pitch. Yousuf Pathan had by then swooped on the ball and rattled the stumps.
But Jayasuriya and Sangakkara rebuilt the innings with a 118-run stand for the second wicket.
Apart from Zaheer, however, India´s bowlers struggled with line and length to help the pair get back on track.
The two were rarely troubled and accumulated runs at a slow but steady pace.
Sangakkara then fell six runs short of a half-century, the wicketkeeper-batsman stepping out to hit left-arm spinner Pragyan Ojha over the top but failing to clear Suresh Raina at mid-on.
Jayasuriya, who was later adjudged the man-of-the-match, went on to reach a century becoming the oldest player at 39 years and 212 days, to achieve the mark.
The knock lacked his customary whirlwind pace, but it put the hosts back on track.
He also went past the 13,000-run mark in one-day internationals, becoming only the second player behind Tendulkar to reach that milestone.
The left-hander added 51 runs for the third wicket with Thilina Kandamby (17) but fell one over after enforcing the batting powerplay.
Farveez Maharoof boosted the score with a quickfire 35, but it proved insufficient in the end.
Ishant Sharma was India´s most successful bowler, finishing with a haul of three for 51 from 10 overs.
Related story
R Madhavan, Himanshi Khurrana hail action against teenager who...
