Gloucester would have gone top of the Premiership had they won but they have still to beat Bath in 15 attempts away from home since English league rugby began in 1987.
Grewcock and Borthwick were at the heart of Bath's mighty efforts, with Grewcock especially producing an irresistible case for England selection against Australia at Twickenham next month.
Bath skipper Borthwick has not even made England's 30-man autumn squad, but Gloucester could not cope with him or Grewcock during a full-blooded tussle.
"They are world-class locks, and I wouldn't want to swap those two for anyone else," said rugby director Connolly.
"Borthwick and Grewcock were strong, and our line-out was exceptional. Both players combined very well, and Grewcock has really worked hard at running with the ball in hand.
"We had to dig it out. It was a case of working hard and stopping Gloucester from getting momentum, which we did."
At Franklin's Gardens, Bruce Reihana's touch-line conversion nine minutes into stoppage time snatched Northampton a dramatic 25-23 victory over London Irish.
The New Zealand full-back produced a virtuoso performance that inspired the Saints to only their second win of the season with two tries and 20 points.
London Irish led by five points as the clock ticked down but a late tackle from Exiles fly-half Riki Flutey handed Northampton one last chance.
Reihana took the quick tap penalty and the ball was spread quickly to Ben Cohen who jinked through the broken defence for his third try in four games.
That drew the scores level at 23-23 and left Reihana with the conversion from wide out on the left touchline to earn Northampton the victory.