Edinburgh were 13-6 ahead minutes before half-time, but the Welsh side levelled with a try before the break and dominated from that point onwards.
The home replacements made a decisive impact as the likes of Olly Cracknell, Gareth Thomas, Aaron Jarvis and try-scorer Scott Otten helped tilt the match the Welsh side's way.
Tandy said: "Over the past three games our bench has been outstanding.
"The boys had been buzzing all week after the high of beating Munster away the previous Sunday, but with just a five-day turnaround it caught up with some of them in their legs.
"We needed a bit more energy because the week had taken its toll.
"We've put in a 23-man shift in the games against Glasgow, Munster and Edinburgh, showing the depth we are starting to grow and what the development arm of the Ospreys is doing.
"It's pretty big for us to back up the positive performance we produced in Munster.
"We had just a five-day turnaround and with all the boys we have missing through injuries and international commitments, it's a real positive focus.
"It would be easy to look at it in terms of the negatives from the first half. We contributed to our own problems then and the referee got on top of us, but that was on the back of a lot of our mistakes.
"But we turned it around and to have only a 12 per cent turnover rate after the break shows how we controlled the game."
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