Crewe boss Steve Davis admitted his side got lucky after Bradden Inman's late goal sank Bury 1-0.
Inman netted his second goal in two games when he poked in Luke Murphy's pass with three minutes to go.
Davis said: "I am pleased to pick up the points in a game where they will feel hard done by as they deserved something from it. When the performances aren't up the mark the hard work gets you through.
"We got a late break and our best player on the day won it for us, and in close games like that it is that little bit of extra quality that gets you the result. We couldn't produce the football we did last week but we are never going to dominate games in this league and there are no easy games.
"We have reached the 33-point barrier which is a good return for a tough half of the season. You don't want to be down the bottom half struggling and we have worked hard to get in this position. That work has got to carry on and we have got to build on it."
The visitors could have taken a fifth-minute lead but Troy Hewitt turned his shot inches over when he beat his man to Matt Doherty's low cross.
Bury's best chance of the half fell to Tom Hopper who couldn't direct his header on target after David Worral picked him out with a pinpoint cross. Worral then had Crewe keeper Alan Martin at full stretch to tip another shot over the bar.
In the second half Martin performed more heroics for Crewe, turning Hewitt's goalbound header over before diving low to keep out former Alex midfielder Steven Schumacher.
Boss Kevin Blackwell, who cannot add to his squad after the club was this week put under a transfer embargo by the Football League, said: "With the chances we have had I can't believe we haven't won the game, let alone lost it.
"We were fantastic but all the good work we put in was wasted when we let ourselves get mugged in the last minute. It is obvious where the problems are. As a manager I have tightened things up and we look a decent side but we need to make changes - but I can't bring anyone in."