Roque Santa Cruz slotted home on 21 after a long throw-in fell into his path in the 18-yard box.
But after David Bentley’s 30-yard strike was tipped over the bar by Tomasz Kuszczak, the Red Devils laid siege on the Rovers goal.
Cristiano Ronaldo hit the upright from 20 yards on 64, and Brad Friedel produced two stunning saves to deny Wayne Rooney on 78 and Tevez’s fierce half-volley on 79.
When Friedel denied John O’Shea with a stunning one-handed save, Rovers seemed destined to win.
But the defending champions got their equaliser with two minutes left when Tevez headed home after Paul Scholes had flicked a corner on.
The draw means United are three points ahead of Chelsea.