After an early rain delay at the start of Saturday's schedule, Anderson topped Frenchman Gilles Simon, 6-3, 6-7 (6-8), 6-3, in the first semifinal. Murray was even at 3-3 in his matchup against Viktor Troicki when the rain returned and halted play for the day.
Murray won the Queen's title in 2009, 2011 and 2013. Another victory would make the Scotsman the fifth player to win this title four times, joining John McEnroe, Boris Becker, Andy Roddick and Lleyton Hewitt.
Anderson, meanwhile, reached his 10th career ATP final and will try for his third title. He hasn't won a tournament in three years, last capturing a crown at Delray Beach in 2012.
The South African has lost six finals since, including a setback to Kei Nishikori earlier this year in Memphis.
Anderson had a chance to win in straight sets on Saturday, but the seventh- seeded Simon saved a match point at 5-6 in the tiebreaker. The Frenchman was broken in the eighth game of the third set and Anderson converted his third match point to finally end the match.