The Scot, who still refers to the Ulsterman as "the Gaffer" after their time together at Celtic, was a happy man after seeing Villa claim just their second Barclays Premier League win of the season at the Stadium of Light. In the process, he piled further pressure on O'Neill, whose team have now won only one of its last 17 league games.
However, Lambert said: "It's a tough, tough job, it really is a tough job. But if there's one person who will turn it around, it will be him. I have got the utmost respect for the Gaffer. I am not in a position to sit here and say I have got one over him or anything. It's a game of football I had to win and Aston Villa had to win, and we did it. It's a good three points."
In a contest between two sides who have found the going difficult in recent weeks, one goal was always likely to be enough to claim the points, and it was Villa who snatched the advantage 12 minutes after the break.
Christian Benteke, who had caused problems all afternoon, climbed above Phil Bardsley at the back post to head down Matt Lowton's cross and Gabriel Agbonlahor pounced to volley into the roof of the net from close range.
It was scarcely more than the visitors deserved after largely containing their hosts throughout, and it was telling that the man who came closest to registering for the Black Cats was skipper Lee Cattermole, who has never scored for the club.
Cattermole's 84th minute effort was Sunderland's only serious attempt on goal on another frustrating afternoon for O'Neill's men.
Lambert, who had to withdraw left-back Joe Bennett after just 11 minutes and then saw replacement Eric Lichaj limp off 19 minutes from time, said: "I thought we deserved it. I thought we were excellent today considering the knockbacks we took with the left-backs and Ashley [Westwood] making his debut. I thought we were excellent."
O'Neill was still able to take positives despite hearing his players booed from the pitch for the second time in five days, but admitted the responsibility for finding a solution lies with him.
He said: "It's me, I'm the manager, I'm the manager of the football club, absolutely. Yes, it's difficult, it's been a difficult week for us, obviously, losing against Middlesbrough and then not scoring here. It's been a difficult week, but we will come through. We threw just about everything at them to try to get on the scoresheet, but it eluded us today again. But it will turn."