The defender fired over an open goal from a yard out after Conor Coady hit the post.
Emi Martinez also denied Coady in the second half at Villa Park as the hosts drew just their fourth game of the Premier League season.
It kept Villa ninth in the Premier League, while Wolves remain 12th with both sides now in danger of seeing their seasons fade away in the final 10 games.
Defeat at Sheffield United in midweek dented Villa's surprise European hopes and they tried to recover quickly.
Watkins had already hit the woodwork more times than any player in the top flight this season and he struck it for a seventh time when his 25-yard rocket rattled the crossbar after four minutes.
Inconsistent Wolves – with three wins in their previous 14 top-flight outings – struggled to match Villa's tempo as they pressed for an opener.
Pedro Neto tried to be proactive on the rare occasion he saw the ball but was quickly shut down by a home defence who had clearly marked him down for special treatment.
Despite Villa's momentum, though, they never tested Rui Patricio during a first half which, the brief flashpoints aside, often lacked spark.
Watkins' shot deflected wide three minutes into the second half but from the resulting corner, Neto fashioned the visitors' first chance.
Coady headed clear and Neto was left racing towards goal with just Matt Targett covering.
He forced the forward wide but Martinez still had to parry his shot behind.
It was the cue for Wolves to up their game and Saiss somehow missed his golden chance after 57 minutes.
A neat short corner saw Neto cross for an unmarked Coady at the far post and the defender – who scored his first Premier League goal against Manchester City on Tuesday – planted a diving header against the post.
The ball bounced across goal but a stretching Saiss managed to lift the ball over from almost under the bar.
It was a huge let off for Villa who had dropped from their first-half levels to allow Wolves to wrestle back control.
Nuno Espirito Santo's side were chasing a winner and Martinez bailed Villa out with 14 minutes remaining.
Ross Barkley inexplicably played the ball back into the box after Joao Moutinho's free-kick and Willian Jose's shot deflected to Coady five yards out.
He had just Martinez to beat but the goalkeeper raced out to turn his shot wide from point-blank range.
But Villa almost stole it in injury time when Patricio's low save denied Watkins and Konsa lashed the rebound wide.