Once Villa went ahead through Ramsey, it looked like they would run away with the game but they did not make the most of their chances.
The hosts had the ball in the back of the net again two minutes later through Morgan Rogers but the offside flag was raised.
Watkins then sent a header wide while Youri Tielemans' strike from 20 yards whistled past the post.
Villa manager Unai Emery, who was watching on from the stands after picking up three yellow cards, was concerned when Tyrone Mings went down injured with half an hour played.
A collision with West Ham's Mohammed Kudus left Mings clutching his left knee in agony, just three months on from his return to action from an anterior cruciate ligament injury, but he came out and sat on the bench for the second half.
Watkins posted on Instagram after the game: "I've had some serious knee injuries, this isn't one of them."
While Villa will be disappointed not to win, they are unbeaten in 11 home games, with only Arsenal (14) on a longer such run in the league.