Lammens arrived from Royal Antwerp for £18.1M as he replaced Andre Onana between the sticks for the Red Devils.
The 23-year-old kept yet another clean sheet on Saturday as he held off Sunderland in what was an intense clash for United who are looking to maintain their spot inside the top 4.
The United goalkeeper denied Noah Sadiki one-on-one before he made an important save in the second half against Brian Brobbey.
Speaking after the clash, Hargreaves revealed that he believes the young goalkeeper could be named signing of the season over the likes of Bryan Mbuemo and Matheus Cunha.
“I’ll tell you who’s probably been the best signing of all of them: Senne Lammens,’ Hargreaves told TNT Sports. ‘What did he cost? Not a lot so he’s been value for money.
“I think United looked at it and thought, “what gets us up the table and gets us into the Champions League places?” It’s not a new midfield player or a goalkeeper, it’s goals.
“So they added Cunha and Mbeumo, guys with Premier League experience who can get you 20-odd goals and United are third right now so I think that’s been proven correct.
“But I do think if they hadn't had sorted the goalkeeper out, they wouldn’t be in the top-four.”
Lammens said his side need to push on next season after feeling “proud” of his debut campaign as he helped the side back to the Champions League.
He said: “We’ve been growing, and now it’s the next step. Next season is going to be a big one. So everyone is going to step up again.
“Because this season was okay, but it’s not our end goal. So we have to keep building on this and try to get better.”