Antonio rose highest at the back post to head in Gokhan Tore's cross and decide a game of few chances.
The goal came eight minutes after Bournemouth's Harry Arter was sent off for a second bookable offence.
Cherries goalkeeper Artur Boruc saved well from Enner Valencia in the first half, while Adrian denied Joshua King.
A club-record 56,977 supporters were in attendance as West Ham played their first league match at their new home, having left Upton Park at the end of last season.
As the players came on to the pitch, the traditional bubbles were blown in to the air and their club song played over the tannoy.
For the most part the home supporters were fairly quiet, though, with the vast arena meaning they are not as close to the pitch.
In fact, the noise levels were so low the travelling Bournemouth contingent chanted "is this the Emirates?" in reference to the atmosphere at Arsenal's stadium.
But having seen their team concede late in Monday's league defeat by Chelsea and Thursday's Europa League qualifying draw with Astra Giurgiu, the Hammers fans had a late goal of their own to cheer when Antonio nodded in.
Bournemouth's start to the season will concern manager Eddie Howe. The Cherries have lost their opening two games to slip to the bottom of the table.
They created very little against West Ham and tested goalkeeper Adrian just twice through King - the second in stoppage time when the Spaniard got down well to save.
Club-record signing Jordon Ibe showed flashes of his talent with his direct running, but fluffed a big chance in the second half when he squared the ball rather than shooting when in front goal.
Once midfielder Arter was dismissed - his first yellow card for dissent towards the assistant referee, and second for dragging back Cheikhou Kouyate - Bournemouth dropped closer to their own area and were soon punished.
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West Ham boss Slaven Bilic: "It was difficult, we knew it would be. The guys did brilliantly. Bournemouth were better than us in the first 15 minutes of the second half but they didn't hurt us.
"We just deserved it. We were pushing and wanted it more. We're delighted to end the first game at this beautiful stadium with a win and a clean sheet."
Bournemouth boss Eddie Howe: "It was a tight game with little goalmouth action. It was going to swing on a moment of brilliance or a mistake. Unfortunately the mistake for us was going down to 10 men.
"We've been told about dissent - it's been drilled into us. If we lose our discipline we could have a flurry of red cards. Unfortunately we fell victim to that. The first booking has really cost us."
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West Ham host Romanian side Astra Giurgiu in the second leg of their Europa League qualifying play-off match on Thursday (kick-off 19:45 BST). The match is tied 1-1 after the first leg.
Bournemouth travel to Morecambe in the EFL Cup on Tuesday at 19:45 BST.
Match ends, West Ham United 1, Bournemouth 0.
Second Half ends, West Ham United 1, Bournemouth 0.
Attempt missed. Jonathan Calleri (West Ham United) right footed shot from the centre of the box is close, but misses to the left. Assisted by Mark Noble following a fast break.
Dan Gosling (Bournemouth) wins a free kick in the defensive half.
Foul by Mark Noble (West Ham United).
Foul by Simon Francis (Bournemouth).
Ashley Fletcher (West Ham United) wins a free kick in the defensive half.
Attempt blocked. Simon Francis (Bournemouth) right footed shot from the centre of the box is blocked. Assisted by Dan Gosling.
Attempt saved. Joshua King (Bournemouth) left footed shot from the centre of the box is saved in the centre of the goal. Assisted by Steve Cook.
Dan Gosling (Bournemouth) wins a free kick in the defensive half.
Foul by Mark Noble (West Ham United).
Pedro Obiang (West Ham United) is shown the yellow card for a bad foul.
Adam Smith (Bournemouth) wins a free kick on the right wing.
Foul by Pedro Obiang (West Ham United).
Substitution, West Ham United. Pedro Obiang replaces Gökhan Töre.
Corner,  Bournemouth. Conceded by Winston Reid.
Substitution, Bournemouth. Dan Gosling replaces Ryan Fraser.
Goal!  West Ham United 1, Bournemouth 0. Michail Antonio (West Ham United) header from very close range to the centre of the goal. Assisted by Gökhan Töre with a cross.
Attempt missed. Jonathan Calleri (West Ham United) right footed shot from the centre of the box is too high. Assisted by Ashley Fletcher.
Substitution, West Ham United. Ashley Fletcher replaces Håvard Nordtveit.
Andrew Surman (Bournemouth) wins a free kick in the defensive half.
Foul by Winston Reid (West Ham United).
Corner,  West Ham United. Conceded by Lewis Grabban.
Substitution, Bournemouth. Nathan Aké replaces Jordon Ibe.
Second yellow card to Harry Arter (Bournemouth) for a bad foul.
Foul by Harry Arter (Bournemouth).
Cheikhou Kouyaté (West Ham United) wins a free kick on the right wing.
Andrew Surman (Bournemouth) wins a free kick in the defensive half.
Foul by Gökhan Töre (West Ham United).
Substitution, Bournemouth. Lewis Grabban replaces Callum Wilson.
Foul by Joshua King (Bournemouth).
Mark Noble (West Ham United) wins a free kick in the attacking half.
Delay over. They are ready to continue.
Delay in match Cheikhou Kouyaté (West Ham United) because of an injury.
Jordon Ibe (Bournemouth) wins a free kick on the right wing.
Foul by Mark Noble (West Ham United).
Foul by Andrew Surman (Bournemouth).
Mark Noble (West Ham United) wins a free kick in the defensive half.
Attempt saved. Jordon Ibe (Bournemouth) left footed shot from outside the box is saved in the centre of the goal. Assisted by Harry Arter.
Substitution, West Ham United. Jonathan Calleri replaces Enner Valencia.
Michail Antonio scored five minutes from time to give West Ham victory over Bournemouth in the first Premier League match at the London Stadium.