With South Africa’s World Cup winning scrum-half Faf De Klerk still to return to action, Sale badly lacked tempo during the opening 40 minutes.
Steve Diamond rang the changes early after the interval and got a response as Coenie Oosthuizen crossed over.
Warriors failed to build on their tally but were relieved as they held on for victory.
As well as De Klerk, Sharks were also missing fellow Springbok Lood De Jager as well as England’s beaten finalists Tom Curry and Mark Wilson.
The hosts on other hand were almost at full strength, with Scotland lock Jonny Gray the only one of the 19 men they sent to Japan yet to return to club duty and it showed as the Warriors bossed the early exchanges.
With Sharks at least doing well to shut down attacking channels, Hastings went for an ambitious drop goal attempt but the ball sliced off his foot and drifted wide.
His next effort off the tee did fly over as he nudged Warriors ahead on the quarter-hour mark.
But it was a shaky kicking display from the Scotland fly-half as he again skewed past the poles with another penalty after Jake Cooper-Woolley was penalised for handling in the maul.
Glasgow were well on top but a three-point advantage was little reward for their efforts, so Hastings made certain to split the posts when he was handed another chance after an illegal Byron McGuigan tackle on 27 minutes.
The thermometer was sitting just a couple of degrees above zero but Sale looked frozen stiff with the ball in hand.
And they were little better as Glasgow broke through for the opening try four minutes before the break.
Fraser Brown’s line-out was plucked by Ryan Wilson before Sam Johnson turned up the speed levels as he flicked a George Horne pass round the corner for Van Der Merwe and there was no catching the Canada international as he darted for the corner.
Sale were hoping the change of ends would produce a change in fortunes.
They got a stroke of luck when a cruel bounce denied Van Der Merwe a second try as he chased after a clever Hastings kick to the corner.
And they finally built up the momentum that got them on the scoreboard on 63 minutes.
Diamond waited just a couple of minutes after half-time before replacing his front-row and it was substitute prop Oosthuizen who squeezed over the line for the score that drew Sharks back within six points of Rennie’s team.
The Sale replacements had made a huge difference, giving them the upper hand at the set-piece and Glasgow’s nerves appeared to be jangling as the game moved into the final 10 minutes.
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