Anderson suffered a second-round exit at Alexandra Palace last year and was made to battle by England's Hunt, who is ranked 84th in the world.
'The Flying Scotsman' showcased early quality, hitting the Shanghai 120 finish on each 20 to force a deciding leg, before wrapping up the set with a 96 on a clinical double 18.
Both players held throw with Anderson knocking in a second Shanghai of the match, but Hunt broke through to make it 1-1 with a 72 checkout on tops.
Another deciding leg was required in set three after Hunt finished a 54 on tops following an Anderson bounce-out, which would have wrapped up the frame.
Hunt converted to lead the match but Anderson locked back in, making it 2-2 in sets courtesy of a 14-dart leg and a classy 88 bull finish, recording a 108.17 average.
Hunt edged ahead in the deciding fifth, but Anderson surged back, breaking throw with a 70 checkout on tops before hitting double three for a hard-fought win.
"I started off great but then up here (in my head) is a different thing," Anderson told Sky Sports.
"The first game is the hardest and I couldn't get into a rhythm.
"One minute I was alright, then all over the place, but I'm through. It gets the heart pumping (with the crowd)."
'Goldfinger' Gilding impresses
Earlier, a ruthless start from Andrew Gilding helped him get past 22-year-old Cam Crabtree 3-1 in an encounter which saw him squander seven match darts.
'Goldfinger' swept the first set and upped the gears to take out 116, then completed a 12-dart leg on tops to move 2-0 up.
Crabtree woke up in the third, tallying his first leg of the match and pinning double five in the decider to take the set after weathering Gilding's huge 161 out.
Gilding reset in set four, positioning double 20 with Crabtree out of the picture to seal the win with his eighth overall attempt.
England's Luke Woodhouse defied some errant finishing to win nine of the last 12 legs and down burly Croatian Boris Krcmar 3-1.
Krcmar edged a first-set decider with a superb 121 finish via the bull, before 'Woody' made it 1-1 by whitewashing the second, which peaked with a 110 checkout.
Woodhouse – the 25th seed – survived six set darts to steal the third, then claimed the all-important fourth to book his place in round two.
Welsh debutant David Davies set up a second-round clash against defending champion Luke Littler, winning a nervy opening match 3-0 over Belgium's Mario Vandenbogaerde.
In the evening session, 2024 world champion Luke Humphries takes on fellow Englishman Ted Evetts to begin his campaign.
Saturday's 2026 PDC World Darts Championship results
Afternoon session
Mario Vandenbogaerde 0-3 David Davies
Andrew Gilding 3-1 Cam Crabtree
Luke Woodhouse 3-1 Boris Krcmar
Evening session