The point was enough to take Villa out of the Premier League's bottom three as Lambert watched on from the stands because of his one-match touchline ban.
Lambert said: "We've played the two Manchester clubs and Arsenal in back-to-back games and it's a tough thing mentally.
"But I thought we played really well. We were excellent and might have nicked it. We defended really strongly. We are playing really well.
"We are a young side which is growing together and the way we are playing, we are a threat."
Lambert defended his decision not to include fit-again striker Darren Bent amongst the 18 players selected to face Arsenal.
He said: "It is my choice. I thought the squad I went with could win us the game. I've got 25-26 players to think about.
"The most important thing is this football club and I pick a team and squad I think will do well."