A resounding defeat at the Emirates Stadium consigned Villa to their third consecutive league loss and left them two places and three points above the relegation zone.
To compound their misery it was the sixth straight league match in which Lambert's team have failed to score.
The chief executive backed the Scottish manager to arrest the team's alarming slump this week but after taking only 12 points from their last 18 league games, the pressure is mounting at Villa Park.
Lambert, though, is willing to fight for his job and the club's Premier League survival and called on his players to do the same.
"You probably can't print the things I'm feeling. Negatives, yes, I don't like being in this position, we've been in this position for too long," Lambert told a news conference.
"Since I've been here it's always been fight, fight, but you've got to keep going, you don't throw the towel in, you keep going. There's going to be twists and turns throughout the season.
"I thought for 60 minutes of that game we were well in it, the disappointing thing for me was the acceptance to lose more goals.
"It was a strange game, I didn't think it was a 5-0 game, but it was and you have to take it. It wasn't good enough for us to lose by that margin."
Villa are 16th in the table with 22 points and host leaders Chelsea next Saturday.
(Reporting by Sam Holden, editing by Mark Meadows)
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