Grigg's brace added to Adam Forshaw's strike to maintain the Bees' unbeaten start to the season and inflict a first league defeat on new United boss David Weir.
Forshaw's goal came with 10minutes before the break when he was allowed space on the edge of the penalty area to curl a right-footed effort past goalkeeper George Long.
A frenetic second half started with Conor McAleny going close to doubling Brentford's advantage, before Blades defender Neil Collins headed home a corner to equalize.
Grigg restored the hosts' lead shortly afterwards, firing a pin-point McAleny cross beyond Long.
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- Adam Forshaw