Both nations would have been worthy winners and whoever faces France in Monday's final will have to play out of their skins to deny them a second World Cup win.
Both goalkeepers made superb and crucial first-half saves with France's Hugo Lloris diving full length to turn away Tottenham team-mate Toby Alderweireld's shot on the turn, before Thibaut Courtois denied Benjamin Pavard with his legs.
Umtiti's goal was a cracker, as he outjumped Marouane Fellaini to score his third and easily most important international goal from Griezmann's pinpoint delivery.
But it was a brilliant defensive display from France who kept the tournament's top scorers goal-less, providing the bedrock from which they earned victory.
Villain
I can't find one today and wouldn't want to. This was just a sublime game of football, well worthy of the occasion.
Stat chat
Samuel Umtiti's winner was the 32nd headed goal at this World Cup, the most since 2002 when there were 35.
Only Harry Kane (six goals) has been involved in more goals at this World Cup than Antoine Griezmann (three goals, two assists).
There have now been 158 goals at this World Cup, 48 of which (30%) have come from set-pieces (not including penalties). In the previous six World Cups combined, 26% of all the goals came from set-pieces.
Belgium have never kept a clean sheet in their 14 World Cup knockout games.
French striker Olivier Giroud has now played nearly eight hours of football (464 minutes) at the 2018 World Cup without managing a single shot on target.
Belgian playmaker Eden Hazard completed six dribbles in the first half while the other 21 players completed eight dribbles between them. He also played 23 passes, all of which found a team-mate.
They said what?
Jurgen Klinsmann, former Germany head coach, now pundit: "Marouane Fellaini is intimidating. In the warm-up, you smell each other, looking how they are doing. If you look at him, you think I don't want to deal with him. The body language, the presence is what Fellaini has. You do not want physical contact with him."
Piers Morgan, English talk-host: "High quality game but I've seen nothing that should terrify England if we make the final. It's coming home."
Did you know?
When French goalscorer Samuel Umtiti first moved to Barcelona, his mother went with him and always made him finish his homework before he could go to training.
What's next?
The second semifinal is tomorrow morning, with England meeting Croatia in Moscow at 6am (NZT).
