The England striker converted a penalty in the 87th minute after Fabio had been fouled by Steven Naismith to guarantee United a top-two finish in Group C.
The visitors had done most of the attacking at Ibrox, with Rooney central to many of their moves, but had been denied by wasteful finishing and good goalkeeping.
United needs a point against Valencia, which is three points behind in second, to seal top spot in the group. The defeat eliminates Rangers, which instead goes into the Europa League.
Tottenham also reached the second round of the Champions League in its first season in the tournament with a 3-0 win over Werder Bremen.
Younes Kaboul and Luka Modric scored in the first half at White Hart Lane and Peter Crouch added the third in the 79th minute to secure progress even before the final group match at FC Twente on Dec. 7.
A place among the final 16 teams will further boost Tottenham's profile and finances. Teams in last season's tournament were awarded an additional €3 million on top of what they had already earned just for making it past the group stage.
Even with Rafael van der Vaart out injured and Jermain Defoe on the bench until the 57th minute, Tottenham had too much firepower for a Bremen side that was eliminated with the loss.
