Bryan Craig's men also secured a try bonus point thanks to scores from Colin Stuart, Callum Champion, James Ferguson, and Kyle Adams.
The junior Ferguson brother and Nick Bell added the rest of the points from the kicking tee, as the home side secured a first win of the season.
The match didn't start particularly well for the hosts, though, who were behind inside 5 minutes following a try from Lasswade winger David Hall. Hennessy's conversion and a further score from Niall Gray just moments later put the Reds in a tricky place at 0-12 after as many minutes.
However, the home side responded and were buoyed by a quick-fire strike as the versatile Colin Stuart, again part of the second row in the absence of the injured Ross Martin, crossed the line just a minute later.
Ferguson couldn't convert but added a penalty shortly after to further reduce the arrears.
In what was a match of two evenly-poised sides, there would be no more scoring for nearly half an hour.
When points did come, however, it was try time for the hosts again as the recalled Kyle Adams found his way over the line, before powerful number eight Champion powered through to all but seal the deal with just over 10 minutes to play.
Ferguson and replacement Nick Bell added both conversions to give the hosts some breathing space.
The match was made safe when that man Ferguson crossed the whitewash to secure the bonus point, and Bell's conversion and late penalty meant that Ali Elliot's late score for Lasswade was no more than consolation.
There were real positives from the encounter, namely the inclusion of Scotland RL cap Callum Cockburn on his return to the club, and the performance of Ferguson in his less-preferred position at stand-off.
A tough test awaits next weekend as the club face the much-fancied Howe of Fife, but a 5-point victory sets them up nicely for the trip across the Forth Bridge.
Full-time: Haddington 32-24 Lasswade