Dan Biggar kicked 11 points and Andrew Bishop crossed for a late try to put daylight between the two sides at the end but the Welsh outfit were forced to battle for every point.
But after the resignations of director of rugby Scott Johnson and head coach Sean Holley in midweek, Tandy admitted he was relieved to launch the new Ospreys era on a positive note.
"I'm glad to get off to a winning start, if pretty frustrated with a lot of the game," he said.
"I felt it was stop-start, we couldn't really get a flow to it and there were too many turnovers but all in all, I'm pleased to get a positive result and a first win.
"After the week the boys have had, they responded well in training.
"Backing up the result in Edinburgh last weekend is a good thing but we'll need to improve on this over the next few weeks.
"We had a good start with an early penalty try but I thought that when we got into the redzone, when we were looking to add a few more tries, there were a couple of infringements picked up that just took the momentum out of the game."
Alun Wyn Jones returned from a toe injury but Tandy was still without 16 players due to international commitments.
"With the week we've all had it was important that we got the win," he added.
"It was a fairly young team out there and that has to be a positive.
"A lot of these guys were playing when we had the unbeaten run at the start of the season so we know what they can do."
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