Having brought with them a strong team, the Top 14 title-holders continued their run of conceding just one try per game in 2014/15, while offering plenty of threat in attack.
The Saints started brightly enough, setting up repeated scrums on the visitors' line. But it took until the last play of the game for them to finally cross the whitewash, George Pisi's spectacular finish bringing Franklin's Gardens to its feet.
Glenn Dickson converted, adding to Stephen Myler's two earlier penalties to seal the 13-3 win, the first time that a match between the Saints and Castres had been settled by more than seven points in over two years.
But while they didn't get the tries they wanted, Wood believes that the team can be pleased with the win.
"We always want to push for four tries and go for five points," he told SaintsTV. "We knew in the back of our minds that that was what it was going to take if we were going to put pressure on Leinster.
"But we were under no illusions. We were up against the Top 14 champions who brought a big heavy pack. There was speculation during the week about whether they'd bring a second team given that they're out of the competition, and away from home French teams can sometimes lose focus.
"But they came here to play and picked a pretty much full strength side. They gave us a real game. They were really stubborn in defence, came looking for the big hits and we had to front up.
"We took scrums early on with the view that we would grind them down and hope that some gaps would open up later on, but fair play to them, they're not Top 14 champions without good reason.
"They stuck with us all the way and it was just pleasing to get the win, never mind the four tries."
The Saints stay in cup action for the next fortnight, with Saracens coming to Franklin's Gardens on Saturday, February 1st for a game which is likely to decide the fate of both teams in their chase for the LV= Cup semi-finals.