Winger Nicky Walker, signed from Shaw Lane, pounced four minutes into injury-time to end a run of three matches without a win — and earn Sinnott his first three points in his second match in charge.
Fifteen minutes earlier man-of-the-match Nathan Stainfield had gone close for Trinity in a match that looked destined to finish goalless before Walker’s winner.
Sinnott said: “We were looking at what we could do to get us over the line and whether players were tiring.
“But I thought Nicky had got one last moment in him and it was a great time to score.
“You always hope you are going to get one more opportunity and he finished it off very well.
“There is a lot more to come from Nicky. He has got blistering pace.”
Henrich Ravas, who has signed a deal to extend his loan from Derby County for the rest of the season, made three early saved against Harriers.
At the other end Walker had a good effort on goal.
Ashley Worsfold also threatened for the Holy Blues in an even start.
Ravas then denied James Pearson as the side fighting relegation matched the play-off chasers.
Craig King and Brandon Fleming were both yellow carded around the half-hour as Trinity fought hard for their new manager.
Winger Walker again went close for the home side just before half-time.
Five minutes into the second half Ravas repeated his good work of the first half to keep out a Harriers’ free kick — and made further important stops around the hour mark .
Bradley Beatson’s important deflection kept the scoreline goalless when Trinity came under more pressure in the second period.
In the 70th minute the home keeper came to the rescue in a goalmouth scramble.
A minute from time Ravas punched clear a corner and then saved again — stops that proved vital when Walker struck.