United went into the game without Wayne Rooney - out with a hamstring injury - and lost Javier Hernandez early on after he was tackled by Jonathan Woodgate, but took the lead in the 27th minute when Nani played a one-two with Darren Fletcher before clipping the ball past Asmir Begovic.
Red Devils goalkeeper David de Gea then pulled off great stops to deny Andy Wilkinson and Jon Walters, but the Spaniard had looked shaky dealing with crosses and it came as little surprise that the hosts' 52nd-minute equaliser came from a corner, which Crouch, recruited from Tottenham for a record £10million last month, met with his head to open his account for his new club.
The result means United leapfrog Manchester City to return to the top of the table, but they are only ahead of their derby rivals, who earlier beat Everton 2-0, on goal difference.
It was also just deserts for Stoke, who put in a battling performance and have now lost only once in 12 games in all competitions this term.
A trip to the Britannia Stadium had always promised to give a searching examination of De Gea's ability under the high ball and he looked insecure in the opening seconds, flapping at Rory Delap's first long throw into the box before the danger passed.
Moments later the Potters survived a scare of their own as Hernandez found himself through on goal.
Woodgate made a challenge from behind which sent the Mexican sprawling into the onrushing Begovic, but while Hernandez stayed down hurt and United boss Sir Alex Ferguson cried out for a penalty, referee Peter Walton allowed play to go on.
Glenn Whelan was then booked for upending Patrice Evra before Phil Jones saw his header from Anderson's free-kick cleared by Delap.
Hernandez was unable to continue and in the 11th minute came off for Michael Owen, scorer of two goals in the midweek Carling Cup win at Leeds.
Another Delap throw led to Stoke winning a corner and De Gea looked stranded when it came in from Matthew Etherington, with Crouch heading wide at the back post but penalised for climbing over Jones.
At the other end, Dimitar Berbatov - who had been called in as Rooney's replacement to make his first league start of the season - nodded Anderson's corner off target.
The deadlock was broken just before the half-hour mark when Nani linked up with Fletcher, burst into the area and fired past Begovic.
Within seconds, Stoke nearly equalised as De Gea did superbly to tip Wilkinson's fierce drive onto the woodwork.
Ponderous play from Begovic almost let in Nani to make it 2-0 but the Portugal international put his effort over the bar.
Back came Stoke, and after Delap had cracked a shot wide from distance De Gea produced another fantastic save to turn Walters' shot around the post.
Woodgate headed over from the resulting corner with De Gea again failing to convince in an aerial situation, and Crouch and Jermaine Pennant then both slid in to no avail in an attempt to get on the end of Walters' cross.
Having finished the first half strongly, Stoke picked up where they left off after the restart and seven minutes in, were level.
Etherington swung in a corner and with De Gea stood on his line, Crouch rose to head home.
Berbatov sent a shot into Begovic's arms, but it was Stoke who had all the momentum and Crouch looked set to score again when he brought down Pennant's chip in front of goal, only for De Gea to get in the way enough for the ball to deflect over.
Begovic then got down to save a low effort from Anderson and De Gea punched away Marc Wilson's firm free-kick.
After Ashley Young's strike had been batted away by Begovic, the former Aston Villa winger came off for Ryan Giggs while Danny Welbeck replaced Berbatov.
Giggs delivered a ball into the six-yard box, but Ryan Shawcross was on hand to get there ahead of Owen, who then looped a header over the bar.
Both sides had chances to win it towards the end, with Crouch sending a header and volley wide before Giggs fired over in stoppage time.