Kyle, on-loan from Coventry, headed in early on and the home defence held out for their first clean sheet of the season as Brighton battled to come back.
With a strong wind behind them, Hartlepool applied pressure from the start and were ahead after 12 minutes.
James Brown's shot from Joel Porter's pass was brilliantly pushed wide by goalkeeper Michael Kuipers but when the corner was taken short on the right, Brown crossed and Kyle was at the far post to head in.
Brighton had to defend in depth as the wind helped Hartlepool pile on the pressure and when they broke the Seagulls saw on-loan Robbie Savage screw his shot well wide from the edge of the area after good work by Joe Anyinsah.
Anyinsah hit the home bar with a shot that swerved in the wind after 62 minutes while Richie Jones was only inches away from doubling the home side's lead at the end of a good move.