Notts came into Wednesday's (AEST) contest fresh from Kewell's maiden win as manager this past weekend and wasted little time asserting their authority over the visitors.
Kane Hemmings helped the home side in front less than 10 minutes into the match and John Stead soon doubled their advantage heading into half-time.
While Crawley came out determined in the second half, later buoyed by Robert Milsom's red card, they were unable to prevent Nathan Thomas from securing Notts all three points, with Dominic Poleon's late effort proving mere consolation.
With Notts now in the midst of a four-match unbeaten run, it appears Kewell has turned the corner at the club.
"I’m very happy with the result but more happy with the performance," Kewell said.
"We stuck in there in moments when the tide was turning a little bit but then we pounced and looked compact and steady.
"Crawley had one or two shots from distance but one in 50 of those go in.
"I’m proud of the players because it’s been a tough week. We’ve got back-to-back wins now and we can push on."
Kewell's men will next face Macclesfield Town on Sunday (AEST) and the 40-year-old expects nothing less than maximum effort from his players once again.
"We’ve started something, now we have to continue it," he said. "Macclesfield drew 3-3 at Newport tonight and it will be a tough game - we have to be prepared for that.
"We’ll enjoy this moment but, come tomorrow, we focus on Macclesfield."
