Roma had led through Daniel Osvaldo’s first-half strike but having failed to turn possession into pressure, they were hit with a late blow.
The result means neither side have yet won a match this season but took them both above champions AC Milan, who now sit just two places above the relegation zone, where Inter Milan find themselves.
The final whistle was met with a cacophony of jeers from disgruntled home fans.
Enrique admitted he was concerned about his team’s results.
“The assessment cannot be positive because of the result but the team did everything we expected in terms of our gameplan,” he said.
“But we didn’t take enough risks to finish off our chances. Siena played very well in the second half while we struggled to move the ball around and create chances.
“There isn’t a specific problem that worries me, everything worries me.
“In the second half we didn’t do what we should have done, there was a bit of fear, which is normal because we haven’t won yet.
“We’re a long way from what we want but I remain confident because there’s another game in three days.”
Siena coach Giuseppe Sannino was delighted with his team.
“I’m happy with the performance, we played against a big team and put them in difficulty,” he said.
Roma started by seeing a lot of the ball without creating anything, while Siena repeatedly looked dangerous on the counter.
Inside the first quarter of an hour Emanuele Calaio had a shot deflected behind off Simon Kjaer while Franco Brienza’s header from Angelo’s cross was brilliant tipped away by Bogdan Lobont.
And yet the hosts went in front on 25 minutes as Jose Angel crossed from the left and Marco Borriello took the ball to the byline before firing it across goal where Osvaldo was waiting at the back stick for a simple tap in.
However, it was still Siena who looked the more likely to score as Calaio dragged a shot just beyond the far post.
In the second period a stinging shot from Osvaldo brought Zeljko Brkic into action to punch the ball clear.
Midway through the period Francesco Totti got his weight behind a free-kick but Brkic got down to save comfortably.
Kjaer then came to Roma’S rescue as he spread his body on the line to block a Pablo Gonzalez header from close range after Calaio headed the ball back into the danger zone.
Miralem Pjanic set his sights from distance for the hosts but fired just beyond the angle of post and crossbar.
But then came Siena’s sucker punch two minutes from time as Brienza’s speculative effort crashed back off the post and Vitiello was on hand to side-foot home the rebound for a deserved equaliser, the club’s first goal this season.
Osvaldo had a late chance to snatch a winner but headed wide.