The two-time defending champions trailed by 18 points in the second quarter, but with Green driving them they produced another dominant third period.
No team has come back from a 0-3 deficit to win a best-of-seven playoff series. The Trail Blazers will try to fend off elimination when they host Game Four tomorrow (HK time).
Green had 20 points, 13 rebounds and 12 assists, keeping the Warriors afloat in the first half before Stephen Curry came alive with 21 of his 36 points after the interval.
"He was like a wrecking ball out there, destroying everything in his path," coach Steve Kerr said of Green.
Klay Thompson added 19 points for the Warriors, who were again without injured star Kevin Durant.
CJ McCollum led Portland with 23 points. Damian Lillard, relentlessly double-teamed by the Warriors, added 19.