If there are more than one of those games sort it by highest scoring to least scoring.
The Vikings defeated the Rams by a score of 23-21 on 1989-11-05, with seven field goals and a safety (not sure if you specifically wanted field goals only, or just no touchdowns). I've seen this game referred to as the most points scored by a team without a touchdown.
I'm sure there are other games (I vaguely recall watching a 6-3 game once), but I am not aware of a definitive list.
Searching for "most field goals in a game" uncovered an article referencing the above game as one of several involving 7 field goals (the record, 8, was in a game with a pair of touchdowns as well). In two of those games, the field goals accounted for all of the scoring for the winning team.
- Bengals defeat the Ravens, 21-7, 2007-11-11
- Cowboys defeat the Packers, 21-6, 1996-11-18