Murray, who was installed as fourth seed at the All England Club, crashed out of the AEGON Championships in the opening round last week.
And in only his second match on grass before SW19 begins next Monday, the 25-year-old was beaten 6-7 (5/7) 6-4 10-7 by Tipsarevic amid the grand surroundings of Stoke Park.
Murray, however, will be grateful for a challenging afternoon against the world number eight from Serbia as he fine-tunes his build up to Wimbledon.
Several lively rallies lit up the third game and Tipsarevic was pushed hard on his serve before holding.
The opening set continued to be keenly contested with both players' serves proving impenetrable and a tie break was needed to separate them.
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Twice Murray trailed with Tipsarevic serving at 5-4 up, but twice the Scot fought back as he steadied his nerve.
Finally making inroads with his serve, he clinched the first set by outmanoeuvring Tipsarevic before firing a simple forehand.
Murray fired a stunning forehand early in the second set only to be broken in the fifth game, Tipsarevic somehow getting his racket to a bullet serve and the Scot then finding the net.
Two break points arrived in a hard-fought 10th but Murray was unable to capitalise on either, ushering in a tie break to decide the match with the rivals locked at one set each.
Murray showed his frustration as he was beaten by an easy backhand to trail 6-4 and he continued to be outgunned as Tipsarevic prevailed in the first to 10 decider.