Michael Jacobs' header gave the hosts a third-minute lead but the Alex stuck to their task and it was only an injury-time strike from Leigh Griffiths that finally sealed all three points.
"We had a lot of chances to put the game away and we didn't take them," said Jackett, whose side had taken only one point from their previous three Sky Bet League One matches.
"But there were some pleasing aspects. I was pleased to have created as many chances as we did, particularly in the second-half when we did work their goalkeeper.
"We always needed that second goal and it was a long time coming but when it did it sealed the win.
"And it was a good clean sheet because Crewe were everything that I expected.
"It was a big day for them and it was a situation where they could express themselves without not too much pressure on them.
"We had to work very hard and all the way through the second half I was looking for that second goal. I always felt that we needed it."
Second-placed Wolves now face a huge top-of-the-table table clash against leadres Leyton Orient at Molineux on Sunday.
Jackett added: "We are really looking forward to it.
"Leyton Orient have goals in their side, which is a major asset, and I am sure on Sunday the top two will produce a very good day and hopefully it will work well for us."