Ireland knew maximum points in their final Pool A match at Fukuoka Hakatanomori Stadium on Saturday would see them through at the expense of Japan or Scotland, who are set to meet on Sunday unless the decisive match is cancelled due to Typhoon Hagibis.
Joe Schmidt's side had the bonus point secured just before half-time, captain Rory Best and Tadhg Furlong crossing before the excellent Sexton scored a brace of tries.
Aki became the first Ireland player to be sent off in a World Cup match for a dangerous tackle on Ulupano Seuteni but it did not prove to be costly.
Jordan Larmour, CJ Stander and Andrew Conway added further tries in the second half, with Sexton nailing four of five conversion attempts before his replacement Joey Carbery was on target twice.
Ireland hold a two-point lead over Japan and will face New Zealand or South Africa in the last eight next weekend, while Samoa bow out with a heavy defeat and could only muster a solitary try from Jack Lam in the first half.
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Best gave Ireland the start they were looking for when he touched down after four minutes as Samoa were forced back by a driving maul and they soon had a numerical advantage with Seilala Lam sin-binned for a hit on Jacob Stockdale.
Furlong ploughed through tackles for a second five-pointer inside 10 minutes and Sexton exchanged passes with Larmour before rounding off a slick move with a well-worked try.
Sexton successfully converted for a third time, but Ireland were mistaken if they thought they would have things all their own way as Jack Lam forced his way over to get the Pacific Islanders on the board.
Aki was given his marching orders with 29 minutes gone when his shoulder caught Seuteni in the face, yet Sexton got the bonus point in the bag just before the break by capitalising on slack defending to cross in the corner.
It was not evident that Schmidt's side were a man down as Conor Murray whipped a pass out for the onrushing Larmour to score wide on the right, Sexton again adding the extras.
Referee Nic Berry sent TJ Ioane to the sin bin for not rolling away as Ireland dominated, Stander crashing over and Conway getting on the end of Carberry's kick to claim try number seven.
Sexton and Murray get Ireland ticking
There were concerns over Sexton's fitness after he was replaced at half-time in the victory over Russia, having missed the defeat to Japan with a thigh injury.
The World Rugby Player of the Year allayed those fears with a superb all-round performance, controlling the tempo of the game along with the influential Murray, kicking superbly and showing his eye for a try.
Murray was as composed as ever, combining brilliantly with Sexton and also displaying his quality with ball in hand and the boot before he was replaced just over 10 minutes into the second half.
Aki a big loss, but Henshaw return a boost
Aki will be facing a ban after his dismissal and the barnstorming centre will be a big loss. On a positive note for Schmidt, fellow centre Robbie Henshaw made a timely return and came through his first match of the tournament after recovering from a hamstring injury.
Photo Getty Images Caption: Ireland fly-half Johnny Sexton scores his second try