The Hammers saw off French outfit Lyon, while the Gers had to rely on extra time to get past Portugal’s SC Braga.
Both British clubs will now face German opposition in the final four.
Here’s everything you need to know about the latter stages of the competition.
Which British clubs are still in the Europa League?
With the likes of Leicester City and Celtic meeting their their demise in the group stage of the competition, only West Ham and Rangers remain as British representatives in this year’s Europa League.
The Hammers put three past Lyon without reply in an assured away performance on Thursday evening, with goals from Craig Dawson, Declan Rice, and Jarrod Bowen settling the tie after a 1-1 draw in the first leg.
Scottish champions Rangers had to overcome a one-goal deficit from their first lef trip to Portugal, but did so in style against Braga at Ibrox.
The Gers went two goals ahead, and could have had a hatful more on another day, before being pegged back late on by their visitors.
With the scores tied at 2-2 on aggregate and no away goals to consider, the match went into extra time. Kemar Roofe’s 101st minute strike ultimately proved to be the difference, securing a first European semi-final for the Glaswegian club in 14 years.
Who will West Ham and Rangers meet in the semi-final?
An all-British semi-final clash was avoided when the draw was made last month.
Instead, both clubs will come up against Bundesliga opposition.
Eintracht Frankfurt arguably scored the result of the quarter-finals after sealing an emphatic 3-2 victory over tournament favourites Barcelona.
The German club raced into a three-goal lead before fending off a late rally from the hosts to pick up a 4-3 aggregate triumph.
Their reward for an inspired display in Spain is a clash against West Ham
For their part, Rangers will face RB Leipzig, who beat Atalanta on Thursday.
After a tight 1-1 draw in the first leg, a Christopher Nkunku brace was ultimately enough to see off the Serie A side.
When will the semi-final ties take place?
Both semi-final first legs are set to take place on Thursday 28th April with the second leg following seven days later. Both games will kick-off at 8pm (GMT).
West Ham will host Eintracht Frankfurt in the first leg, while Rangers will travel to RB Leipzig, with a return fixture at Ibrox the following week.
Where will this season’s Europa League final take place?
This year’s Europa League final will be held at the home of the competition’s most successful club.
Six-time winners Sevilla may have seen their bid for a seventh Europa League ended by West Ham - but they will still play a part in the final as it will take place at their Ramon Sanchez-Pizjuan Stadium home.
That game will take place on Wednesday 18th May with an 8pm kick-off (GMT).
Who are favourites to win this year’s Europa League?
The exit of big names Barcelona and Atalanta have blown this year’s competition wide open, according to bookmakers
RB Leipzig are now favourites at 11/10, followed closely by West Ham at 15/8.
Behind them, Eintracht Frankfurt come in at 5/1, while Rangers remain rank outsiders at 13/2.