Three penalties apiece from Nick Macleod and Olly Barkley meant the scores were finely poised at 9-9 with eight minutes left in Friday night's Aviva Premiership match at Edgeley Park.
However, Sharks full-back Rob Miller latched on to a loose ball to touch down and Macleod converted to lift them one place to sixth, two points ahead of Gloucester.
Diamond was particularly thrilled with the effort, especially after a difficult three weeks with the sacking of head coach Tony Hanks.
He said: "We're delighted with the win, we've worked hard since Saracens and we were unlucky to lose the game against Newcastle Falcons away.
"We've done a lot on our defence and it showed in the last 15 minutes that they've been listening to what we've been doing in training.
"At this end of the season, if we can scrape into Europe then they're the four points that might help us get there."
Their narrow victory now sets up a mouth-watering clash away at Gloucester. Assuming the Cherry and Whites defeat bottom club Newcastle at Kingsholm tomorrow, the winner of that match will probably decide who qualifies for Europe.
Diamond said of Gloucester: "They're a very well-coached team. To go there and get a result would be great. We've got clarity in how we want to play at the moment and not to play too much rugby in our own half, but take opportunities as Rob Miller did."
Meanwhile, Bath first-team coach Brad Davis admits their chances of qualifying for the Heineken Cup are "highly unlikely".
The west country outfit currently reside in eighth position, five points behind Sale and, if they are to play in Europe's top-tier next season, results would need to significantly go in their favour.
"It's not mathematically out of the question so we will look at how the results go over the weekend," he said.
"We can't fault the effort and commitment of the guys, they were absolutely outstanding. We went for the full 80 minutes which is pleasing.
"We asked for a response after the Northampton game and I thought we got that in bucketloads. On reflection, a draw was probably a fair result.
"The try they scored was a bounce of a ball and a hack through, and sometimes you get on the right end of them and sometimes you get on the wrong end of them."
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