The game was settled with just 15 minutes played when Gabriel Agbonlahor, who scored the only goal in a 1-0 win in the corresponding fixture last season, capitalised on Lee Cattermole's error to round goalkeeper Vito Mannone and roll the ball into the empty net.
Sunderland thought they had levelled seven minutes after the restart when Emanuele Giaccherini fired home from Steven Fletcher's knock-down, but a harsh offside flag ended his celebrations.
The home side were otherwise toothless in their attempts to restore parity, and Villa substitute Marc Albrighton was unable to take advantage of either of the two late opportunities with which he was presented to cement victory after Phil Bardsley had cleared Christian Benteke's shot off the line.
On the controversial incident, Lambert said: "If it's offside, then it can't be a goal, can it? I couldn't see - it's hard to see from where I'm standing, but the best one in the place to see is probably the linesman."
Lambert was understandably delighted with the result, Villa's first league win in six attempts, as well as the returns of Ron Vlaar and Benteke from injury.
He said: "I thought we deserved it - I thought we could have run out more comfortable winners. We should have at least had three or four, the chances we had.
"Coming up here was always going to be tough because of Sunderland's situation, but I felt the Swansea game gave us a platform to try to build.
"But we are a football club that can't carry big injuries to big players, and I think it was evident here, the two lads coming back gave us a major lift."
Lambert hopes to add to his squad this month, but he was giving nothing away,adding: "If I can, I will, if I can. Yes, there are ones I would like to try to bring in. But if we can, we can; if we don't, then we don't."