The visitors took an early lead when Elliot Daly scored with a penalty after only two minutes.
Leicester soon hit back as Twelvetrees scored a kick of his own to get his free-scoring afternoon up and running.
The first try of the match was scored by Wasps as Nic Berry touched down a minute later. However, Daly was unable to make the conversion.
Twelvetrees kicked over another penalty, before he converted his own try with half an hour played to give the hosts some breathing space.
Ryan Davis's uncoverted try cut the deficit for the away side, but another penalty from Twelvetrees on the stroke of half time gave the Tigers a 16-11 lead at the break.
Two further penalties from Twelvetrees put Leicester further in front before his converted try in the final ten minutes sealed the win.