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Steven Gerrard's side have not dropped points in 12 games since they last faced Hibernian in September.
They had to fight hard to maintain that perfect streak before Hagi's second goal in successive games broke the deadlock at Ibrox.
Rangers are now 19 points clear of second-placed Celtic, who have four games in hand and face Hamilton later on Saturday.
Bidding to stop Celtic winning 10 Scottish league titles in a row, Rangers are in pole position as the new year looms.
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Rangers quickly dominated possession and threatened when Connor Goldson met James Tavernier's corner with a header that went straight at Hibs keeper Dillon Barnes.
Hagi shifted the ball onto his right foot before unleashing a shot which squirmed wide just as Ryan Porteous slid in.
But there was no-one keeping tabs on the Romanian a moment later as Rangers took a 33rd minute lead.
Glen Kamara fed Kemar Roofe in the box, with the former Leeds forward's cross sneaking through Porteous' legs to find Hagi rushing past a sluggish Paul Hanlon to hook home.
Hibs had a penalty appeal turned down when Hagi's high boot brushed Joe Newell's face when the Rangers player tried to clear a bouncing ball in the box.
That close call seemed to jolt Rangers back into life and Roofe fed Ryan Kent but Paul McGinn produced a vital last-ditch tackle, while Barnes made another solid stop to deny Balogun.
Hibs were throwing all they had at the league leaders as the game ticked into the final stages.
Allan McGregor had to sprawl down to his left to push away Melker Hallberg's swerving shot before Borna Barisic got an important toe in to deny Kevin Nisbet on the rebound.
smg/gj