Caleb Gaylard pulled one back on the stroke of halftime from a cornerkick.
Centreback Aaron Jones made it 3-1 midway in the second spell.
"They [Sox] had some chances in the first half but not many in the second," said Hastings, who adopted a 4-1-4-1 formation rather than a flat 4-4 one.
"They hardly touched the ball for the first 25 minutes."
Red Sox coach Aaron Clegg said the Rovers' first goal was "legitimate" but the hosts dominated the half in terms of creating clear-cut opportunities.
Gaylard, Clegg said, had kicked a freekick from their right flank in the 30th minute with the ball hitting the 'D' inside the net before bouncing back into play.
"The linesman [ref's assistant] signalled a goal but the ref, who didn't see it, waved play on."
From the ensuing play, the "blatantly offside" Rovers had scored on the other end for a 2-0 lead, instead.
"They [Rovers] had more possession but we were shaping up behind the ball and they were not threatening or penetrating," Clegg said, adding Lawson's 25m goal was a "speculator" devoid of any "creative play".
He also claimed Jones' goal had deflected off his leg after Red Sox failed to clear from a cornerkick.
"We lost our structure from the time it was 3-1 and we played too long too quick."
In the other Chatham Cup match in Napier on Saturday, Your Solutions Taradale lost 3-1 to Palmerston North Marist in the battle of two Federation League sides.
All the goals came in the first half with Thomas Mosquera scoring twice and Jordan Martens once.
Daniel Ball slotted a penalty for the hosts.
"We weren't outplayed. We struggled in the first half at different periods but our application was much better than against Red Sox and our attitude was fantastic," Taradale coach Chris McIvor said of his men, who went down 6-2 to the Sox in a Fed League clash in Palmy North the previous Saturday.
"In cup ties it comes down to key moments and we didn't take them," he said, lauding Napier Boys' High School pupil, James Harris, 17, for making his debut in the midfield and ensuring the soccer clubs in the Bay are able to build an ongoing rapport with schools.
