Jarvis, on loan from Norwich, had hit a hat-trick at Millwall in midweek and followed up with a double blast after defender Justin Miller had put Orient ahead.
Top-scorer Chris Greenacre had cut the deficit after the home side had raced into a two-goal lead, but it was not enough to boost Tranmere's play-off hopes.
The visitors had started brightly and after nine minutes Calvin Zola flashed a header wide, but it was Orient who suddenly found their form.
Wayne Corden should have put the home side ahead after an impressive build up down the right, then Shane Tudor sent in a cross but no-one could find a finishing touch.
But Orient broke the deadlock on 18 minutes when Tranmere could only half clear Matt Lockwood's left-wing corner and Miller fired home through a ruck of players.
As Orient kept their foot on the gas, Jarvis just failed to turn home a Tudor cross and then had a penalty shout turned down.
But the on-loan striker gained his reward on 29 minutes when he burst into the box and fired home from six yards following Gary Alexander's excellent left-wing run and cross.
Tranmere responded and cut the deficit when Danny Harrison whipped in a cross from the right and John Mullin headed back across the goal for Greenacre to slot home from eight yards.
But Orient quickly regained their two-goal cushion when some neat play down the left ended with Lockwood's smart cross being turned home by the alert Jarvis.
After the break Orient continued to press and gave the makeshift Tranmere defence a thorough examination.
Jarvis had a great chance to complete back-to-back hat-tricks early in the second half when Lockwood supplied a delicious cross from the left but the striker screwed his shot wide from six yards.
The 20-year-old then hit a tame effort straight at Rovers keeper Gavin Ward.
Greenacre went close to pulling one back for the visitors, who had plenty of possession but failed to provide a potent threat to the Orient goal, and it was the home side who should have stretched their advantage with Alexander and Corden both going close.