Venable's clout in the fourth inning put the Padres up 5-0 and two more runs in the seventh gave San Diego a 7-1 advantage.
Logan Forsythe went 2-for-2 with a solo homer en route to the victory.
Jason Marquis (8-2) stretched his unbeaten streak to eight straight starts, limiting the Braves to two runs on five hits over 7 2/3 innings, overcoming five walks while fanning four.
Thayer, however, fanned Justin Upton for the final out, recording his first save as the Padres took the opener of a three-game set.
Heyward also hit a bases-empty homer in the eighth for National League East-leading Atlanta.
Braves starter Julio Teheran (4-3) was roughed up for five runs on six hits over six full innings.
Swim Week Miami Beach 2014 [PHOTOS]