Rene Howe's penalty secured the points for the Gulls, who had to withstand plenty of pressure from the home side, but Exeter failed to turn their second-half dominance into goals.
"I'm delighted and I've got a smile on my face," said Taylor, who captained Exeter to the old Fourth Division championship in 1990.
"It was a solid performance all-round and the players did the team proud.
"We got the goal and knew it would be tough in the second half, especially with such a strong wind. People may question whether we deserved the win, but we battled well in the second half.
"Getting three points here is massive for us and it is great for the fans as well.
"We still had some good opportunities in the second half to get a second goal but we did weather a bit of a storm second half but we weathered it admirably.
"We haven't been playing too badly, that result will stand us in good stead heading into the Cheltenham game on Saturday which will be tough.
"I haven't spoken to Martin yet, but I spoke to him a couple of hours before the game and he is at home. We don't know the extent of his illness, but I'm sure he will be happy."