Weaver (5-0) logged his seventh straight start without a loss, stifling the Twins on three hits, walking two and striking out two.
The Angels right-hander improved to 7-2 for his career against Minnesota but notched his first victory at Target Field.
Alberto Callaspo lined a two-run homer to left field in third inning and added a run-scoring single while Albert Pujols contributed a pair of RBI for the Angels, who took the opener of a three-game series in Minneapolis.
Minnesota starter Francisco Liriano (0-5) remained winless over seven starts this season, surrendering four runs on five hits in a five-inning effort.
2014: The Year in Music [PHOTOS]