The Villans had conceded 15 and scored none in their previous three games and Wayne Routledge added to their woes as he fired the hosts ahead in the ninth minute.
But relegation-threatened Villa had looked set to take the points when Andreas Weimann levelled just before the interval and Christian Benteke scored from the spot when Nathan Dyer conceded an 84th-minute penalty for a foul on Weimann.
However, Graham fired home a rebound in a late goalmouth scramble, to leave Lambert with his head in his hands.
A point at least ends Villa's miserable run of defeats and, despite the crushing disappointment, Lambert feels he can take great heart from a gutsy effort by a side with an average age of just 23 years and 119 days - making it the youngest the club had fielded in a Premier League fixture.
He said: "It is a huge point although we should have seen it through. But it's a massive point and we were very good and threatening in the second half.
"In the first half they were coming at us from all angles and we had to weather that storm and we needed a couple of good saves from Brad [Guzan] and they also hit the woodwork.
"But to get the goal before half-time certainly helped us and it had a galvanising effect, it gives you something to get going and in the second half we were very good.
"We have been beaten heavily in the last three games so this is a huge point for the future of the club.
"The young lads in the side have been excellent and when you get beaten heavily you learn a lot about people and it showed they are mentally strong."