Full-back Lachlan Coote scored two tries and kicked two goals as the champions made it eight straight league wins since the resumption to strengthen their position at the top of Super League.
But Saints were well below their best against their bottom-of-the table opponents, who had only 17 players from which to select their team, and needed two tries in the last 20 minutes to finally wear them down.
Woolf said: “We just didn’t approach the game the way we needed to. Their heads were not where they needed to be.
“We weren’t tough enough mentally and errors tend to happen when that occurs.
“Credit goes to Wakefield, who played out of their skins. They completed high and certainly made us work for it. They didn’t go away.
“We were a lot better in the second half than the first.”
Saints were without veteran prop James Graham after he was forced to self-isolate after coming into contact with someone who has tested positive for Covid-19 and they were indebted to a captain’s knock from hooker James Roby.
Woolf added: “Even though he’s not been here long, James has become a real leader and with him we would have been a bit more hard nosed at the start of the game.
“At the end of the day we’ve got to be able to overcome that.
“Robes did what he always does well. He came up with a really good play at the back when we we needed him to.”
Listen to The Final Hooter podcast, featuring exclusive interviews, every week. SUBSCRIBE to Love Rugby League on Apple Podcasts.
More on site