Second half goals from Ronnie Kenneth and Robbie Henderson gave Tayport all three Super League points, at Kelty, with the clincher coming five minutes into stoppage time.
'Port had been unfortunate in a goalless first half, as home 'keeper Alan Fleming made a couple of outstanding stops towards half-time. A one-handed save stopped a cracking Robbie Henderson strike, then, a few minutes later, the same player was denied from close-range, when a goal had seemed certain.
Previously, Stevie McManus hit a shot over the top, after Ronnie Kenneth had knocked down a Mark Whatley cross, while, at the other end, striker Lawrie went close on a few occasions, but did not hit the target.
Neal Ferrie did have to make one crucial stop, after Craig Morrison escaped to go through one-on-one. He could find no way past the Tayport custodian, though.
Two minutes into the second half, a poor clearance from Fleming created an opening for Tayport, with Kenneth getting in behind the defence and sending his low, left-footed effort beyond the 'keeper, into the far corner of the net.
Lawrie then missed a good chance to equalise, at the far post, while Barry Donachie had to throw his body in the way of another Kelty attempt. 'Port continued to defend in resolute fashion, as the hosts began to apply late pressure.
In injury-time, skipper Roberto Morris was forced off, with cramp, but when the game re-started with a wide free kick, Tayport secured the win. Ronnie Kenneth won his aerial challenge at the far post and substitute Ross Lunan's first touch found Robbie Henderson, who rattled the ball home, from a tight angle.