Gary Mulligan netted Gillingham's winner midway through the first-half when he converted Jarvis' cross from close-range for his fourth goal of the season.
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The youngster found fellow striker Delroy Facey who curled the ball just wide to the relief of Jack.
Almost inevitably it was the electric pace of Jarvis that brought Gillingham out of their shell and produced a goal.
In the 23rd minute the winger beat full-back David Worrell to a long ball and crossed for the in-form Mulligan to tap home the opener.
Worrell felt he'd been impeded and showed his contempt for Jarvis with a tackle from behind that brought a deserved booking.
But moments later Jarvis was on his feet and curled a shot narrowly wide following another mazy run.
Rotherham remained a threat and on 31 minutes Facey unleashed a 20-yard effort that had Jack beaten but cleared the crossbar by inches.
The Gills continued to find Jarvis with balls over the top and he again turned the hapless Worrell before firing his shot straight at Neil Cutler.
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And in the 71st minute Hoskins came within inches of an equaliser only for Ben Chorley to deny his close-range effort with a heroic goalline block.
Gillingham should have put the result beyond doubt but both Mark Bentley and Dean McDonald failed to convert glorious chances after Jarvis had played them in.
They were almost made to pay when Eugen Bopp saw a late effort saved before substitute Dave Hibbert's last minute shot was blocked by Chorley.