Arrieta (16-5) allowed a single to Alex Dickerson in the second and Christian Bethancourt's double in the eighth. Arrieta has two no-hitters in the past year.
The Padres brought the go-ahead run to the plate in the ninth, but Aroldis Chapman got the last two outs for his 28th save after Felix Pena was charged with three runs.
Bryant's 33rd home run tied him with Colorado's Nolan Arenado for the NL lead.
Russell smacked his fifth homer in five games, a two-run shot that put the Cubs ahead 4-0 in the fifth.
Arrieta won for the fourth straight start. He had six strikeouts and three walks.
Christian Friedrich (4-10) lost his eighth consecutive decision.
