Victorious in each of their final five league matches last year, the defending champions picked up where they left off as Goretzka struck twice in 12 first-half minutes.
Hoffenheim took a 10-match unbeaten streak into Friday's encounter but were unable to build on Nico Schulz's fine 59th-minute strike in Julian Naglesmann's 100th match at the helm in Germany's top flight.
Robert Lewandowski provided the knockout blow three minutes from the end to trim Borussia Dortmund's advantage back to three points ahead of the league-leaders' tricky trip to fourth-placed RB Leipzig on Saturday.
Bayern began brightly and threatened through Lewandowski and Thomas Muller inside the opening 20 minutes.
Mats Hummels should have gone one better, but failed to poke past goalkeeper Oliver Baumann amid a scramble in the box.
Goretzka eventually made the dominance count with a quickfire double.
He broke the deadlock in the 34th minute with a deflected close-range finish after Baumann saved Lewandowski's initial header, before David Alaba's inch-perfect cross enabled the midfielder to cap a swift counter-attack in first-half stoppage time.
Hoffenheim were more competitive after the interval and it did not take long for Joelinton to draw Manuel Neuer into a smart low save, the follow-up to which Kerem Demirbay dragged wide.
Neuer, though, was given no chance with the next effort as wing-back Schulz speared an unerring drive into the bottom-left corner from 20 yards.
The hosts pressed with renewed hope and an Adam Szalai header drew a crucial denial from the Bayern captain, but the comeback was extinguished when Muller intelligently turned James Rodriguez's dinked pass into Lewandowski's path for the sealer.
Bayern back in the groove
Having lost on their past two Bundesliga visits PreZero Arena, this appeared a tough initiation into the new year for a Bayern side that was less than convincing across the first half of the campaign.
They were certainly made to toil victory but, after a fluent first 45 minutes and second period full of perspiration, Kovac will view this result as a significant step forward in his team's attempt to overhaul BVB.
Goretzka gets among the goals
He has now had half a season to adjust to life with Bayern and two-goal Goretzka appears increasingly comfortable with his surrounds.
The 23-year-old was at his box-to-box best on Friday, bringing industry in the midfield of the pitch and a nose for goal that saw him unleash six shots in total.
