No, they can't. This is because you didn't use an action that allows you to sustain the grab.
The text you've quoted pertaining to sustaining a grab appears to be from the 'Grab' section on page 290 of the Player's Handbook.
That section talks about a specific action called 'Grab' that results in a grab. If you use that action, you can sustain it in such a manner. The brawler didn't use that action, he used one of his powers. That power doesn't have a sustain line, it just ends the grab automatically.
I believe the confusion rises from the PH's use of the indefinite article: "a grab", which makes it seem like this rule applies to all grabs, rather than to just the grabs produced by this action.
This is made clearer in the Rules Compendium on page 243, where 'Grab' is rewritten to explicitly be a power with a sustain line:
Sustain Minor: The grab persists until the end of your next turn.
Notice the use of the definite article: "the grab". This is referring specifically to the grab resulting from using the power 'Grab', not grabs in general.