The Tigers went behind early on but their forwards took control after Frank Murphy crossed on 23 minutes.
Frank Murphy narrowed the gap further and Bath, who had two men sin-binned, could not find the fourth try.
On the half-hour, a promising Connacht move broke down on the Edinburgh 22 with a fumble by scrum-half Frank Murphy.
With eight minutes remaining, Morris gathered McFadden's chip over the top and evaded Frank Murphy's attempted tackle to chalk up the bonus point.
Connacht, with four new recruits in their line-up including half-backs Keatley and Frank Murphy, had failed to cope with facing the wind in the opening half.
Connacht seemed to start dictating play towards half-time but suffered a blow when their skipper Frank Murphy got himself sin-binned for a high tackle on Scarlets full-back Andy Fenby.
Leicester were forced to turn to third-choice scrum-half, Frank Murphy, against Gloucester on Saturday, but the Tigers may have Scott Bemand back from injury in time to face Wasps.
Irishman Frank Murphy, who replaced Ellis against Bristol, is set to start against Gloucester, while 17-year-old academy player Ben Youngs - the son of a former England scrum-half, Nick - could be on the bench.
But Leicester quickly worked their way back into the game and they took the lead through scrum-half Frank Murphy, in for the injured Harry Ellis, who just managed to bundle his way over for a try which was eventually given by the television referee.
Johan Santana, Daniel Murphy, Frank Francisco and Kirk Nieuwenhuis have all missed time due to injury for New York, too.
Niall Ronan, Frank Sheehan and Barry Murphy scored for Munster.
Murphy, Thorstein Pollit, Frank Shostak and others.
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