Ireland knew maximum points in its final Pool A match at Fukuoka Hakatanomori Stadium on Saturday would see it through at the expense of Japan or Scotland, who are set to meet on Sunday unless the decisive match is cancelled because of 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 Rugby World Cup match when he was judged to have dangerously tackled 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 holds a two-point lead over Japan and will face New Zealand or South Africa in the last eight next weekend, while Samoa bows out with a heavy defeat and could only muster a solitary try from Jack Lam in the first half.
Best gave Ireland the start it was looking for when he touched down after four minutes as Samoa was forced back by a driving maul and it soon had a numerical advantage with Seilala Lam set to the sin bin 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 was mistaken if it thought it would have things all its own way as Jack Lam forced his way over to get the Pacific Island side 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 was 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.