Firstly, Guylain Ndumbu-Nsungu cleverly flicked the ball through to Matt Jarvis inside the area, but the youngster's shot went tamely wide.
Moments later, Ndumbu-Nsungu shrugged off Hinds' challenge to release Clint Easton in the area, who saw his first-time shot parried back to him, and then fired the rebound over the bar.
In the fifth minute Alan Pouton delivered a fine through ball to Michael Flynn but, despite having time to size up his shot, the Gills captain fired a poor effort well wide.
As the game settled down, Gillingham continued to enjoy the majority of possession and, after 35 minutes, Jarvis forced a fine save out of Joe Murphy from 12 yards.
The home side were made to pay for that miss when, in the 40th minute, as Hinds rose unmarked at the far post to head home Dave Mulligan's corner.
Gillingham were punished again two minutes later when Cleveland Taylor released Sparrow, who sprinted into the area before calmly placing his shot past a helpless Darren Randolph.
In the 55th minute, Gillingham responded and came close to scoring when Ian Cox's header from Easton's corner was cleared off the line via the post.
Scunthorpe then raced down the field and Dave Mulligan's perfect cross was met by Billy Sharp, but the prolific striker's header missed the target when he should have scored.
After Sharp volleyed another half-chance over, Ndumbu-Nsungu burst into the box at the other end only to see his run blocked. A goalmouth scramble ensued and Gary Mulligan twice saw efforts blocked, much to the frustration of the home fans.
The Iron looked more than capable of extending their lead further, though, and came close on 77 minutes when Ian Morris raced through on goal, only to slip as he pulled the trigger, allowing Randolph to save.
Randolph then produced a wonderful stop to keep out substitute Steve Torpey's header as gaps began appearing in Gillingham's defence.
A late burst of pressure from the home side proved fruitless as Scunthorpe comfortably held on for their first win of the campaign.