Bautista-Agut, seeded fifth for the ATP 250 hard-court event in Moscow, saw off Andrey Kuznetsov 6-3 6-4 to progress to the second round with relative ease.
The Spaniard broke his Russian opponent in game six of the opening set, only for Kuznetsov to hit back immediately.
Bautista-Agut responded in kind, however, before seeing out the set by holding to 30 in game nine.
The world number 16 took control of the second set, as back-to-back breaks saw him move 5-2 to the good.
Kuznetsov rallied and saved two match points to claw back one break in game eight, but he was unable to repeat the trick in the 10th as Bautista-Agut prevailed.
Eighth seed Seppi, who won this event two years ago, had a similarly straightforward passage in overcoming Dusan Lajovic 7-5 6-4.
The Italian finally took advantage of his fifth opportunity to break in game 11 of the opening set, before saving one against his own serve in the next as he took the lead.
A break in game nine of the second set Seppi up for victory, and he wasted little time in holding to love to close out the win.
Elsewhere, Daniel Gimeno-Traver will face third seed Ernests Gulbis in the second round after coming through 7-5 6-3 against Jiri Vesely, and Filip Krajinovic also advanced after a 6-4 6-2 success over Malek Jaziri.