It became very clear on Saturday night in Joe Louis Arena how much the smallest bit of success can change the entire complexion of a team — and possibly a season. In a game the Rangers needed to win in the worst way, what they got was some extra effort where there was none before, the result being a 3-2 overtime win over the Red Wings that sends the Blueshirts home from this brutal nine-game season-opening road trip battered, but no longer beaten. “I think we deserved that one,” said Derick Brassard, the beleaguered center who was called out by coach Alain Vigneault before the game for needing to contribute more, then went out and scored the game-winner on a breakaway in overtime. “It’s not the record we were expecting to have,” Brassard said about his 3-6-0 team, “but we’re going home for a while and we’re going to turn this around.” What might have turned the game around was a moment late in the second period, when the Rangers were down 1-0 after Drew Miller got the Red Wings (6-4-2) on the board early. With the Rangers the power play, forward Benoit Pouliot found a way to bang in a loose puck at the goalmouth and tie the game, 1-1. It was Pouliot’s first goal of the season, his first point as a Ranger, and it enlivened his game. So much so that when the overtime rolled around, there was Pouliot sprawled out on the ice, diving to advance the puck to Brassard, who, after getting denied by the red-hot goalie Jimmy Howard on a breakaway earlier in the game, took a rolling puck and netted the decisive tally. “It’s a big boost, for sure,” Pouliot said about his goal and the energy left in its wake. “It makes you feel good, makes you feel better. In your head, it feels like it’s a big push for yourself.” Just that little bit of boost is what now allows the Rangers to go home for Monday’s game against the Canadiens at a revamped Garden with at least a modicum of momentum. Though they played an assertive game all night, notching a season-high 40 shots on net and 54 total attempts, the game teetered on falling off into another bad-luck loss.