The 22-year-old Republic of Ireland international, arrived at the club on loan from Brighton last summer, and has gone on to make 30 appearances for the club.
Those appearances triggered a clause that Albion had to buy the midfielder - with Molumby now signing a three year contract at The Hawthorns for a fee believed to be in the region of £900,000.
The midfielder insists it was his goal to earn a permanent move when he arrived last summer - but added the players have fallen short of expectations this season and it can;t happen again.
He said: "It was goal at the start of the season to try and get a permanent move here and I’ve managed to achieve it so I’m really happy.
“It’s been a very tough campaign. The expectation this season was to get promoted automatically and if we weren’t able to achieve that, it was all about getting into the play-off places.
“It’s my first year playing senior football where those have been the expectations.
“We’ve fallen short this year and that’s so disappointing. As players, we need to look at ourselves and take responsibility. Ultimately, it’s us that have fallen short.
“Hopefully next season will be very different and we can achieve what the club expects us to achieve."
Molumby made 14 of his 30 appearances from the bench this season and netted in the defeat against Middlesbrough earlier in the year.
He has admitted it has been difficult to get into a rhythym this season - but now his goal is to become a regular starter as Albion prepare for an improved campaign.
He said: “I found it quite tough at times to get a rhythm going, but now that I’m here for the next three years, I want to kick on and make sure I’m a regular in the starting XI.
“I’m determined to improve and prove to everyone that I’m good enough to be a West Bromwich Albion regular.
“I think this is a great place for me grow and improve as a player. I was delighted when Albion came in for me and as I keep saying, it’s a huge club and I want to be a big part of it.
“As players, we need to handle the pressure of playing for a big club like Albion. I’m no different and I need to do more.”