The former Aston Villa and England defender died after suffering a cardiac arrest in April 2017.
He was at Spurs’ training ground at the time, having joined the club as a coach in 2014.
"I'm really proud of the lads," Postecoglou said after a 4-1 win over Aston Villa.
"It was an important day for us today.
"Because it's also the birthday of the late Ugo Ehiogu and he had a big influence on Matty Wells here at the club, I know he played for Villa but he was obviously a coach at our football club.
"Matty spoke really strongly about him. When you hear things like that, and how people affect your life, it helps give clarity to the players about what we're trying to do here.
"Obviously for his wife and his son probably a tough day, but hopefully they get a little bit cheer from the fact Wellsy made it a really important point, that he may be gone but he's not forgotten.
"He just spoke about it during the week. We mentioned him today because it was his birthday today. Like I said, you don't dismiss those kind of things, because he was with our football club when he sadly passed away. So it's important that the memory lives on."