Jamie Short made an early breakthrough for the Peacocks, hitting a belting half volley from distance past Dylan Castanheira after five minutes.
Rodney Marsh was then pushed in the area by Clayton Fortune, earning a penalty which skipper Simon Ingram calmly converted on 20 minutes.
And four minutes later Marsh cut inside to beat Castanheira with a left-footed effort to put Taunton three goals clear at half time.
Marsh was unlucky to see his chip hit the post before Reeko Best slid a shot under the diving Matt Urwin to get Clevedon on the scoreboard after 68 minutes.
With Town going off the boil, the visitors reduced the arrears further on 83 minutes when Best ran through to net past Urwin.
However Taunton held firm to register the three points.