The Cameroonian kept a clean sheet during his maiden competitive appearance for Erik ten Hag's side, who prevailed 1-0 at Old Trafford thanks to Raphael Varane's 76th-minute header.
It wasn't always pretty from the Reds, who were made to work hard for the three points by the visitors in their first match under new head coach Gary O'Neil.
Onana saved all six shots he faced on target, including a smart stop with his feet from Pedro Neto prior to Varane's opener.
That was followed up with two consecutive efforts to thwart Wolves substitute Fabio Silva.
All the good work could have been undone deep into injury time, but referee Simon Hooper waved away Wolves' penalty appeals when Onana collided with Sasa Kaladzjic while coming to claim a cross.