Andy Hessenthaler’s side had to soak up plenty of possession and chances throughout, but Kelly and Palmer both pounced on defensive errors to secure all three points.
The O’s made one change to the side which beat Stevenage 3-0 on Tuesday night, with Jay Simpson, pulling out with a back injury and the O’s only able to name six substitutes.
It was the hosts who started on the front foot and some indecision at the back presented Omar Bogle with a chance for the Mariners, however, O’s goalkeeper Alex Cisak was out quickly to block the shot.
Shortly after Grimsby earned the first corner of the game as James Berrett’s shot was well saved by Cisak, before the O’s cleared the ball behind for a corner, and from the resulting set piece Craig Disley headed wide.
It took Orient a little while to settle into the game and the first opportunity fell to Dean Cox, but he dragged an effort wide of James McKeown’s goal.
Up the other end Omar Bogle was ambitious with a 35 yard strike after spotting Cisak slightly off his line, however, the ball nestled deep into the stand behind the goal where the O’s fans were situated.
Orient were able to force a corner though when a lovely ball through from Nigel Atangana found the run of Bowery and though no-one was able to get on the end of the ball in, the Grimsby clearance thundered off another Mariner and just over the bar.
Cisak was called into action once more as he superbly tipped over Bogle’s 20-yard strike to force another corner for the hosts, which once more came to nothing.
Persistence was key though for the O’s as Bowery never gave up a lost cause, and forced a back pass to be tipped over by McKeown.
As the half wore on it was the hosts who continued to dominate. Danny Andrew’s cross from the left was met by Kayden Jackson at the near post but he just directed the effort over, while just before that Berrett tried his luck from range, though the effort was off target.
Summerfield forced a full-stretch save from Cisak in the last minute of normal time in the first period as Orient will have been relieved to have gone in at the break level.
Grimsby remained on top in the early exchanges of the second period, with Cisak called into action twice in the opening three minutes, pushing Jackson’s shot behind for a corner after the Mariner cut in from the right.
Berrett headed over Chambers’ cross as Grimsby continued to put Orient under early pressure, and the next attack from the hosts saw Chambers fire into the side netting.
O’s first attack of the second half almost resulted in the opening goal as some good skill play from Cornick set Massey free and after coming in off the right saw his shot tipped behind by McKeown.
Orient built on that, and despite Grimsby dominating the game, the visitors broke the deadlock on 57 minutes.
Bowery’s ball in from the right was fumbled by McKeown and Kelly was ready to pounce on the loose ball and slot into an empty net for his first goal in an O’s shirt.
Orient began to mount some continued pressure following the opener as Bowery was almost through on goal, bar a last-ditch intervention from Josh Gowling, and from the resulting corner Yvan Erichot saw an effort deflected behind.
For the hosts Bogle was presented with a glorious opportunity to level when the ball fell to him from a corner, but his turn and shot was wide of Cisak’s goal.
Ollie Palmer was brought on against his former club, and five minutes later he did the customary thing, and score!
It was presented to him after Gowling slipped on the ball and Palmer was ready to pounce. He run forward with the ball before placing it neatly past an onrushing McKeown to double the O’s lead.
The second goal proved to be decisive and from that moment on the O’s remained in control. Grimsby did have a chance to reduce the deficit though in stoppage time as a ball from the right found Chambers, but Cisak was able to smother at his feet.
The hosts did pull a goal back though with two minutes of the four additional minutes on the clock as substitute Dominic Vose’s deflected effort curled past a helpless Cisak to give Grimsby some late hope, though it all proved to no avail as Orient secured their first three points on the road this season.
Grimsby Town: McKeown, Davies, Andrew, Pearson, Gowling, Chambers, Disley, Bogle, Jackson (Vernon 77), Berrett (Vose 69), Summerfield.
Subs: Browne, Mills, Bolarinwa, Boyce, Warrington.
Goals: Vose (90+2)
Booked: Chambers (90+4), Andrew (90+5).
Orient: Cisak, Hunt, Parkes, Erichot, Semedo, Massey, Atangana, Kelly, Cornick, Cox (Palmer 72), Bowery.
Subs: Kennedy, Weir, Sargeant, Pollock, Koroma.
Goals: Kelly (58), Palmer (77)
Booked: Erichot (33), Massey (87)
Referee: Graham Salisbury
Attendance: 4,991 (347 Leyton Orient fans)