Maren Mjelde's second-half spot kick broke the deadlock for the Blues, after Millie Turner was judged to have brought down Fran Kirby inside the penalty area.
The Reds, though, showed plenty of endeavour against the league leaders, and were unfortunate not to draw level late on, as Ella Toone's volley from close range was hooked off the line by the hosts.
Fresh from an international break where United's Mary Earps played in front of a record crowd at Wembley, and Kirsty Hanson received her first senior call-up for Scotland, the Reds travelled to the capital hoping to maintain our five-match winning run.