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The home side twice took the lead on Friday, through defender William Gallas and Dutchman Zenden's stunning strike.
But twice United called on their battling spirit to hit back while manager Alex Ferguson continually prowled the touchline on a night when passions and tempers ran high.
David Beckham curled home their first equaliser midway through the first half before Giggs swivelled and shot home from inside the Chelsea penalty area with 25 minutes remaining to leave them with four points from their opening two league games.
But the visitors, who can be satisfied with surviving a stern test of their credentials by Claudio Ranieri's title hopefuls, left feeling aggrieved that they were robbed of a chance to win the match when referee Graham Poll turned down their strong claims for a penalty.
After 59 minutes, Paul Scholes burst through and, after nudging the ball past Carlo Cudicini, the Chelsea keeper appeared to bring the England midfielder down -- but to United's disgust, Poll waved play on.
Ferguson later remonstrated with Poll, England's FIFA representative, on the touchline after Jimmy Floyd Hasselbaink had been booked for a foul on Giggs, one of five cautions.
The Scot's anger was no doubt stoked by another poor United defensive display, just four days before they attempt to overturn a one-goal deficit in the second leg of their Champions League third qualifying round tie with Hungary's Zalaegerszeg.
Won 1-0 vs West Brom (H)
Lost 1-0 vs Zalaegerszegi (A)
Won 5-0 vs Aarhus (A)
Won 3-0 vs Parma (A)
Lost 1-2 vs Ajax (A)
Won 2-1 vs Valerenga (A)
Won 3-2 vs Bournemouth (A)