Chelsea, who hit the woodwork twice late in the evening game, top the standings with 19 points. United, who started with striker Wayne Rooney on the bench, have now drawn five of their eight league games and are among four teams on 14.
Arsenal, who came from behind to beat Birmingham City 2-1 at the Emirates Stadium, and Tottenham Hotspur, who ended Fulham's unbeaten start with a 2-1 win at Craven Cottage are level with United.
At Old Trafford, United looked to be cruising having taken a 2-0 lead after 25 minutes but Evra's own goal and a rare blunder from goalkeeper Van der Saar, who dropped the ball at the feet of Somen Tchoyi, saw the home side pegged back.
Rooney played in England's goalless draw with Montenegro on Tuesday and said after the match, contrary to the claims of his club manager Sir Alex Ferguson, that he was fully fit.
Fit or not, the striker was among the substitutes against West Brom and Ferguson looked to have got it right when Rooney's replacement, Javier Hernandez, tapped in after goalkeeper Scott Carson dropped a Nani free-kick in the fourth minute.
The Portugal winger then added a second on the break.
West Brom, who won at Arsenal last month, continued to press though and within 10 minutes of the restart they were level.
VAN DER SAAR BLUNDER
United defender Nemanja Vidic was inches away from making it 3-0 when he headed against a post but moments later it was 2-1 courtesy of Evra's own goal.
Then the usually reliable Dutchman Van der Saar, faced with the simplest of catches, let the ball slip from his grasp to Tchoyi who gleefully equalised for the hard-working visitors.
Rooney eventually joined the fray for the last 20 minutes, but played on the left wing and made little impact as West Brom ended the game looking the more dangerous side.
"It has gone from a fantastic first-half performance to giving the game away really, it should have been four or five by half-time," Ferguson told Sky Sports News.
"We've made some glorious chances and it's frustrating we are not killing teams off. It's not acceptable for the supporters or me."
Of Van der Saar's blunder, he said: "Inexplicable, it's a mistake that Edwin probably would make the only time in his life. The last time he made a mistake like that was probably at primary school."
A win for Chelsea would have taken the champions seven-points clear but, without several injured players including forward Didier Drogba, they were below par.
Branislav Ivanovic and Nicolas Anelka hit the woodwork but so did Ciaran Clark for hard-working Villa.
Arsenal fell behind to a Nikola Zigic header, levelled with a Samir Nasri penalty and took the points with Marouane Chamakh's impressive run and shot just after the break.
The win was spoiled at the end with a straight red card for Arsenal midfielder Jack Wilsher
