Griffiths, in only his second game with the club, broke the deadlock as he bundled the ball home, while Burns was able to leave his marker and head in a Roly Bonevacia free kick.
Wellington's wild weather threatened to halt proceedings earlier in the day as puddles were forming on the Hutt Recreation Ground but the rain stopped before kickoff and the pitch drained adequately.
The heavens reopened during the second spell but not with the ferocity of the rain from the middle of the afternoon, which forced the early termination of a curtain raiser featuring a team of former Phoenix players and a local club side.
Wellington and Adelaide came in to the contest within striking distance of the summit of the A-League ladder given the congested nature of the top five teams.
Sydney FC have surged recently, while the front-running Perth have impressed all year and Melbourne Victory are also in the mix. The top six make the playoffs and Melbourne City and Brisbane are jostling for the sixth spot.
Wellington will want to finish inside the top four at worst, with a top-two finish most desirable, as that would give them home advantage for most of the knockout stages.
Phoenix skipper Andrew Durante shook off a calf problem to play today and he marshalled a backline that included the returning Louis Fenton who replaced the suspended Manny Muscat at left back.
Fenton has regularly played at right back under Merrick but was concussed after being struck in the jaw by a ball three weeks ago against Melbourne City.
Playing him at left back seemed somewhat of a gamble by coach Ernie Merrick but it was his assist that provided the goal for Griffiths.
Durante's centre back partner Ben Sigmund was one of the Phoenix's best as he dealt with the pace of Adelaide's strikers and the Phoenix were duly rewarded with a clean sheet.
On a gloomy day in Lower Hutt, the Phoenix shone the brightest and their superb passing game was evident throughout the contest as they proved to be the superior side.
Considering they beat the Victory in Melbourne last week, the Phoenix have to be considered one of the form teams in the A-League and they deserve to be mentioned in the title discussion.
Wellington will head to Perth next week to meet the Glory.
Wellington Phoenix 2 (Joel Griffiths 79, Nathan Burns 82)
Adelaide United 0
HT: 0-0
