Roberto Martinez's side struggled to break down their hosts in Piraeus for 70 minutes until Vertonghen thumped home a long-range effort - only to see Zeca equalise in quick time.
As brief as their celebrations were cut short, Belgium were joyous again as Romelu Lukaku capped a crazy five-minute period of an otherwise dour game by heading home Thomas Meunier's cross to book Belgium's ticket to the tournament.
Vertonghen acknowledged that a tense affair could have gone either way, but was determined to toast Belgium's success.
"The Greeks came out dangerous," he is quoted as saying by Het Nieuwsblad. "They used many long balls, played well on the counter and developed two or three very good chances.
"Our game was not good, but we are qualified.
"We had a slow pace and could hardly find the space. In the second half we got more space and we scored a goal, but it could have fallen on the other side.
"We'll fly back right now, but we're going to have a drink!"
Courtois had to make saves in the first half from Anastasios Donis and Kostas Stafylidis as Belgium failed to make the most of dominating possession and territory.
The Chelsea goalkeeper believes the Red Devils could come up against similarly staunch defences in Russia and must find ways to break through grittier opponents.
"If we had gone behind, it's a completely different story," Courtois added, having also criticised the quality of the pitch.
"We won with character, we kept going until we made it 1-2.
"Of course, the pitch is not the only reason we kept the ball too much instead of playing around.
"When we tried that, immediately there were three or four Greeks on us, we did not get space. We must try to find a solution in Russia against teams that are so strong."