The Premier League side are through to their first European semi-final since they lost the 1976 Cup Winners’ Cup final to Anderlecht after outclassing Lyon in France.
Having drawn 1-1 at home in last week’s first leg despite playing half the game with 10 men, the Hammers took the lead in the 38th minute of the return when Craig Dawson headed in a Pablo Fornals corner at the near post.
Captain Declan Rice slotted in their second goal just before the break, his sidefoot effort from the edge of the area beating Lyon goalkeeper Julian Pollersbeck with the aid of a slight deflection.
The outcome was put beyond doubt when West Ham scored again three minutes after the restart as Fornals sent Jarrod Bowen away to make it 3-0.
Earlier the in-form Christopher Nkunku scored both goals as Leipzig beat Atalanta 2-0 in Italy to win their last-eight tie 3-1 on aggregate.
The Germans had come from behind to draw 1-1 at home in the first leg last week and they went in front in the 18th minute in Bergamo as Konrad Laimer cut the ball back from the right for Nkunku to score.
Leipzig wrapped up the victory with three minutes left as Nkunku converted a penalty he himself had won.
That goal was the 30th of the season in all competitions for Nkunku, who made his full international debut for France last month.
Leipzig, who reached the Champions League semi-finals two years ago, advance to a last-four tie against either Braga or Rangers.
Their second leg went to extra time with Rangers beating Braga 2-1 in 90 minutes in Glasgow after losing 1-0 in the first leg.