Aceves (4-1), making a spot start, limited the Rays to three hits, overcame three walks and struck out two before a 3-hour rain delay brought his outing to an end.
David Ortiz put the Red Sox ahead with a two-run single in the third inning and added a run-scoring hit in the fifth as American League East-leading Boston took the opener of a three-game series.
Tampa Bay starter Chris Archer (1-3) was touched for four runs (three earned) and five hits and five walks over 4 2/3 frames.
Luke Scott drove in the Rays' only run.
Ohio bar shooting arrested, charged with murder
LGBT community has 'bullied the American people': Bachmann