The Warriors grabbed a 3-1 lead in the best-of-seven Western Conference semi-final series.
Curry, coming back after missing the previous four games with a sprained right knee, came off the bench to score 27 of his points in the fourth quarter and extra session. Curry also had nine rebounds and eight assists.
The Warriors trailed 16-2 before the reigning league MVP came off the bench midway through the first quarter.
Klay Thompson scored 23 points and Draymond Green collected 21 points, nine rebounds, seven blocked shots and five assists for the Warriors.
Damian Lillard scored 36 points and handed out 10 assists, CJ McCollum added 24 points, and Al-Farouq Aminu chipped in 18 points and 13 rebounds for the Blazers, who will attempt to extend the series by winning Game 5 is Wednesday night at Oracle Arena.
Curry scored the Warriors' first 12 points of overtime for a 123-118 lead with 1:51 to go. Harrison Barnes' fastbreak layup extended the lead to 125-118 with 1:23 to play.
Lillard made a pair at the line to cut the difference to 125-120 with 1:18 to go, but Curry buried another 3-pointer, and it was 128-120 with 1:05 remaining. The Blazers got no closer than seven points the rest of the way.
Lillard scored 17 first-half points despite making only three of 14 shots from the field, and McCollum added 16 to stake Portland to a 67-57 halftime lead. Curry came off the bench for 11 points for Golden State. Both teams shots poorly -- Portland 39.6 percent and Golden State 39.1 percent.
The Blazers finished at 40.6 percent, while the Warriors improved to close at 46.3 percent.
It was 71-59 when Thompson got hot, sinking four straight 3-pointers as the Warriors went on a 23-7 run to go in front 82-78. They carried an 86-85 edge into the final period.
Golden State led 92-87 with nine minutes left, but McCollum knocked down back-to-back 3-pointers, Aminu buried one from the corner, and suddenly Portland was back on top 96-95. After Lillard and Thompson exchanged treys, Aminu made one of two at the line and it was tied 100-100 with 4:53 remaining.
Curry rained in a trey for a 103-100 Golden State lead, but Lillard converted a driving layup to make it 103-102. Lillard's 30-foot 3-pointer gave Portland back the lead at 105-103 with 2:40 to play.
Lillard sank a pair of free throws to increase the Blazers' advantage to 107-103 with 2:12 to go.
Curry buried another one from beyond the arc, and it was 107-106 with 2:01 to go. Lillard answered with a jumper to make it 109-106 with 1:47 remaining. Green scored on a dunk to close the gap to 109-108, but Mason Plumlee's layup pushed Portland in front 101-98 with 58.6 seconds left.
Harrison Barnes evened the count at 101-101 on a 3-point shot with 51.7 seconds to go.
Lillard missed a fadeaway jumper. The Warriors rebounded and didn't call for a timeout. Curry missed a running bank shot, and Green's tip-in failed to drop as time expired, forcing an extra session.
Portland got out to a 16-2 lead, with Lillard providing nine of the points, as the Warriors started 1-for-12 from the field.
