Coleman's position is under scrutiny after last month's defeats to Macedonia and Serbia led Football Association of Wales chiefs to stall over a two-year contract extension until after the end of the Group A campaign.
Coleman's irritation with the situation has been clear to see and he has demanded the full backing of the FAW if he is to continue, while admitting the association has asked him to change his backroom staff.
But the fourth victory in his 14-match reign, which came thanks to a committed effort from a patched-up side after 11 players withdrew from Coleman's original squad, will have at least alleviated some of the pressure on him.
And the Wales boss said: "Had we lost this game, I would have been out the door, no doubt.
"It's a win and, for me as a manager, looking at our squad of players and all the pull-outs we had, I am delighted.
"I wanted us to look after the performance and we did that. We may have ridden our luck a bit but I thought we deserved to win.
"I enjoyed the win tonight more than any other because there has been a lot said and we were up against it. The players stood up to the test."
Wales could have enjoyed a more comfortable margin of victory had the otherwise excellent Aaron Ramsey not failed to convert a penalty after Church's opener, but they still climbed off the bottom of Group A above Macedonia and Scotland and into fourth place.
But Coleman insists the result will have no influence on any decision he has to make over his future.