Scunthorpe got off to a flying start when midfield man Richard Hinds headed them in front in the 24th minute from a left-wing cross by full-back Lee Ridley.
But they failed to build on that advantage and it was struggling Gillingham who hit back strongly with Gary Mulligan and Jarvis both testing goalkeeper Joe Murphy with good efforts.
The two then combined to bring their side back on level terms in 54th minute when a ball in from Mulligan left Jarvis with a simple tap-in finish.
The real turning point of the game came in the 63rd minute when the visitors had centre-back Ben Chorley sent off for pulling back striker Jermaine Beckford as he broke free through the middle.
Scunthorpe regained the lead from the resulting free-kick which full-back Dave Mulligan curled beyond the reach of goalkeeper Romain Larrieu.
Gillingham were denied an equaliser in the closing stages when a header from substitute Danny Spiller rattled the crossbar and bounced away to safety.
And it was Scunthorpe who completed the scoring in stoppage time at the end when substitute Ian Morris exchanged passes with Beckford before cracking a left-foot shot past Larrieu and into the bottom corner.