But Manchester United's own South American striker rescued a point with just two minutes left on the clock, after goalkeeper Brad Friedel had pulled off a string of sensational saves.
Rovers boss Mark Hughes decided to stick with the same side that suffered a 3-1 defeat at Liverpool six days earlier, while United manager Sir Alex Ferguson made four changes to the side that had secured a crucial victory over Arsenal.
Tomasz Kuszczak replaced the injured Edwin Van der Sar in goal, Nemanja Vidic returned to the side following a spell on the sidelines, while Ryan Giggs and Tevez were both given the nod.
There was early controversy, as Rovers were denied what appeared to be a stonewall penalty with less than five minutes on the clock.
After a mistake by Paul Scholes, Santa Cruz's shot deflected into the path of Jason Roberts. The Grenada international was taken out by Kuszczak, but referee Rob Styles remained unmoved.
Meanwhile, at the other end of the pitch, Cristiano Ronaldo's left-foot strike ricocheted off Ryan Nelsen, but Friedel just got back to claim the ball on the line.
Five minutes later, Michael Carrick found Wayne Rooney in space, the England man centred for Tevez, but Chris Samba made a vital block.
And it proved to be a vital incident as, just past the 20-minute mark, Rovers took a deserved lead. From Morten Gamst Pedersen's long throw, Rio Ferdinand and Vidic got in each other's way and Santa Cruz supplied a cool right-foot finish from ten yards.
As the half-hour mark approached, Ronaldo fired a tame free-kick straight at Friedel, before David Bentley unleashed a stinging shot from fully 25 yards, which Kuszczak had to tip over the bar.
On 37 minutes, United felt they should have been awarded a penalty when Steven Reid appeared to catch Rooney with a wild lunge inside the box, but again the man in the middle waved play on and, six minutes later, Ronaldo's goal-bound header was kept out by Friedel.
Eager for a better showing in the second half, United brought on Nani for Giggs at the break but, in the blustery conditions, neither side was really able to stamp their authority on the game.
On 64 minutes, Reid's poor headed clearance went straight to Ronaldo, but the Portuguese superstar struck the outside of Friedel's far post with a long-range left-foot shot.
Three minutes later, Santa Cruz headed narrowly over the bar from Stephen Warnock's deep free-kick and, seven minutes later, Pedersen came within a whisker of doubling Rovers' advantage after a superb solo run.
Desperate to break the deadlock, Ferguson switched Rooney out to the left and it almost paid an immediate reward. But the visitors came up against a goalkeeper in inspired form.
Friedel twice came to Rovers' rescue on 79 minutes, first from Rooney and then sensationally from Tevez. The big American then denied Rooney in a one-on-one, before keeping out substitute John O'Shea at the back post.
But from the resulting corner, United snatched an equaliser. Nani swung it in, Scholes flicked on and Tevez headed home from close range.