Burns had earlier dropped Adam Hose on one and the former Somerset man made him pay with 36 from 15 balls while Sam Hain (29 off 20) and Ed Pollock (24 off 10) made handy contributions to help Bears stay up with a required rate of more than 10 runs an over.
Earlier, Jason Roy's 74 off 38 balls and meaty hitting from Australian pair Moises Henriques (48) and Aaron Finch (39) had propelled Surrey to a total of 204 for five, the second time they have posted more than 200 in this season's competition.
It was a score that led to injured Surrey batsman Kevin Pietersen tweeting to ask if they could declare.
Surrey were once again indebted to openers Roy and Finch. They had already put together two century stands in this season's Blast and this time added 98 from 55 balls. Finch had to bat with a runner after pulling a calf muscle in the second over but it did not inhibit him too much as struck five fours and two sixes before holing out to long on.
They scored 75 in the powerplay with Roy unleashing some massive hits, with one of his four sixes landing on the balcony of the OCS Stand. He also struck seven boundaries before Jeetan Patel, whose first over went for 17, returned to have him lbw playing across the line.
Surrey had 112 on the board at the halfway stage but could only add a further 71 in the next nine overs as Birmingham's bowlers found some control. The last over cost 21 runs but the visitors then paced their chase to perfection.
* Birmingham will play Glamorgan at 1100 on finals day, with Hampshire v Notts at 1430 and the final at 1845,