The Mariners, playing at this level for the first time since 2010 after winning last term’s National League play-offs, took an early lead through Kayden Jackson’s sixth-minute strike and Ben Davies’ free-kick in the 63rd minute doubled the advantage.
Grimsby had Josh Gowling sent off with 19 minutes left but were able to see out the victory.
Their fellow newly-promoted side Cheltenham secured a point thanks to Billy Waters’ 76th-minute effort as they drew 1-1 at home with Leyton Orient, who had gone ahead via Gavin Massey’s third-minute goal.
Doncaster, relegated from League One, were beaten 3-2 in a thrilling encounter at Accrington.
The visitors blew the opportunity to grab an early lead as John Marquis’ penalty was saved.
Doncaster then saw Romuald Boco put Stanley 1-0 up but swiftly levelled through Tommy Rowe.
Matty Pearson restored Accrington’s lead just prior to the half-hour mark, with Andy Williams then equalising in the 82nd minute before Sean McConville wrapped up all three points for the home side in stoppage time.
It was a happier afternoon for two other teams who have dropped down from the third tier as Blackpool – in Gary Bowyer’s first game in charge – defeated Exeter 2-0 at home courtesy of a Kyle Vassell strike and Troy Brown’s own goal.
Crewe won 2-1 at Stevenage, with Ryan Lowe and Alex Kiwomya netting before Charlie Lee’s reply.
Colchester, the fourth side to have come down, drew 1-1 at Hartlepool after Tom Eastman’s opener for the away team was cancelled out by Billy Paynter’s penalty.
Plymouth, beaten in last term’s League Two play-off-final, suffered a 3-0 home loss to Luton, for whom Danny Hylton, Jack Marriott and Jonathan Smith scored and Dan Potts was sent off in the 89th minute.
There was late drama at Newport as the Welsh side were condemned to a 3-2 defeat by a strike from Mansfield’s Kevan Hurst.
The hosts had earlier twice equalised, with goals from Jack Compton and Joss Labadie replying to efforts from Matt Green and Danny Rose.
Yeovil and Crawley also made winning starts to the season.
Yeovil saw off Notts County 2-0 with goals from Kevin Dawson and Otis Khan, and Crawley were 1-0 victors over Wycombe thanks to a Jimmy Smith strike.
Crawley had Andre Blackman dismissed late on.
Cambridge’s home clash with Barnet finished 1-1.
Piero Mingoia scored the opener for the hosts and Alex Nicholls replied for the visitors.