Four second half goals - including one from 17-year-old debutant Regan Slater - saw the Blades complete the Checkatrade Trophy campaign with a win.
Leon Clarke was also on target on his return from injury, whilst an own goal and a Jack O'Connell knee completed the scoring with all four Blades goals coming at the end of the 197 travelling fans.
Lavery made his first start in a red and white shirt as United completed their involvement in the competition. Youngsters Kelly, Hallam and Slater also started, whilst the bench was made up of Done and Clarke, returning from injury, and Academy products.
In a slow start, United created the first chance but Whiteman's effort was comfortable for Warrington. Grimsby, however, opened the scoring with their first opportunity with 14 minutes gone. Vernon threaded a ball down the left channel and JACKSON beat Long with a shot into the bottom corner.
Long saved as Jackson got clear again and around the half hour Lavery had two chances to open his Blades account but his free-kick was blocked and then he fired over after a couple of stepovers.
On 38 minutes Brown was ideally-positioned and unmarked but headed wide a curling free-kick and moments later Hallam attempted to turn in an O'Connell cross as United forced the hosts back before the break.
On top late in the first half, the Blades kept the pressure on early in the second period. Warrington again saved from Whiteman but one of the Blades youngsters levelled the scores. On 53 minutes, Grimsby half-cleared but SLATER was perfectly-positioned to drill back a shot beyond the goalkeeper's left hand.
Done and Clarke were introduced as they returned from injury and United remained on top, Hallam went close before two goals in four minutes put the visitors firmly in the driving seat.
Firstly, BOYCE turned Clarke's cross past his own goalkeeper and then, from a short corner, a Brooks cross was diverted home by O'CONNELL.
The home side gave themselves hope as the game entered the final ten minutes. A Browne shot forced Long into a save but DISLEY was on hand to tuck home the rebound.
But a fourth goal secured victory for United, CLARKE looked confident as he picked the ball up on the left and beat Warrington with a powerful shot at the near post.
Grimsby: Warrington, Pearson, Chambers (Tuton 76), Disley, Mills, Jackson, Berrett (Bolarinwa 56), Boyce, Vernon (Browne 56), Jones, Comley.
Unused: Andrew, Gowling, Vose, Summerfield.
Blades: Long, O'Connell, Basham 9cummings 76), Lavery (Clarke 63), Brown, Freeman (Done 45), Whiteman, Kelly, Brooks, Hallam, Slater.
Unused: Wright jnr, Gilmour, Perryman, Cantrill.