Sarr gave Rob Edwards' side a 12th-minute lead with a goal reminiscent of David Beckham's memorable strike at Wimbledon in 1996, before Karlan Grant equalised in first-half stoppage time.
But Sarr missed a 73rd-minute penalty in a game controlled by Albion, who should have won at a canter only to have to settle for a point.
Darnell Furlong's fierce volley was tipped over the crossbar by goalkeeper Daniel Bachmann after unconvincing Watford headers from Francisco Sierralta then Christian Kabasele following Jed Wallace's corner.
Bachmann then saved Watford again, this time blocking Grant's shot with his legs after the striker was allowed to control a wayward shot from John Swift.
But the Hawthorns crowd was silenced by Sarr's sublime strike.
Picking the ball up inside his own half from a Craig Cathcart header, Sarr looked up and spotted goalkeeper David Button on the edge of his penalty area and lofted a shot over him and into the empty net.
It was ambitious but fully intentional and Button could only look on in disbelief then shake his head, with the stadium stunned.
Shocked by the goal, West Brom resumed where they left off, however.
Grant toe-poked the ball wide from Wallace's cross, then the striker's angled goalbound shot was deflected wide off Kabasele.
Bachmann showed fine reflexes to keep out Swift's shot diving to his left before Grant headed over the follow-up from Wallace's chip.
West Brom's pressure finally told in the second minute of time added at the end of the first half.
Grant side-footed the ball into the top corner after controlling Grady Diangana's diagonal pass over the top.
West Brom threatened to take the lead shortly after the restart when Conor Townsend cut in from the left and saw his angled drive palmed away by Bachmann.
The onslaught continued and Grant blazed a shot on the turn high over the bar from eight yards after everyone missed a Wallace corner.
But Sarr missed the chance to put Watford ahead for a second time when his poor spot-kick was saved by Button's legs after Semi Ajayi pushed the Watford striker.