It was a tight margin of victory, 14-10, with just two penalties converted in an otherwise scoreless second half, but Redpath was relieved to have taken the points in what was a forwards-dominated encounter.
"We knew it was going to be tough - a three to five-point game - but I thought we were probably the dominant side, particularly in the set-piece," he said afterwards.
"After the six weeks we have had, and we have lost a few, hopefully that can turn around the season for us because we need to push on with two home games (coming up against Connacht and London Wasps)."
Gloucester led 11-0 thanks to two Freddie Burns penalties and a try from James Simpson-Daniel, but captain Gavin Duffy crossed for Connacht before the break.
Redpath added: "I thought we should have gone in 11-0 up at the break, but one mistake from a dropped high ball and Duffy finishes off. Credit to them, they never gave up, but I thought we had strong field position throughout the game and our discipline was outstanding.
"There were some big performances from a lot of people but not necessarily flash - just honest hard-working performances, and we needed that after the six weeks we have had.
"I know Connacht's record has not been good over the last wee while, the same as ours, so any team would have taken that today.
"Not many teams come here and win comfortably. It was a dogfight and you have to show a lot of respect to Connacht for that because they never gave in and made us work hard for everything. So we are delighted."
Connacht boss Eric Elwood said it was a winnable game for his charges, who remain bottom of Pool Six albeit with their first Heineken Cup point now on the board.
"It is difficult because that was a game we felt we had a huge opportunity to get our first win in the Heineken Cup, so now we cannot feel sorry for ourselves," he insisted.
"We know we are capable of mixing with this opposition. Yes, they will be a different proposition on their home patch, but the challenge now for us is, 'Can we do to them next week what they did to us today?"'
The hosts managed to reduce the arrears back to 11-10 early in the second half, but Gloucester's superior set-piece, particularly the scrum, had Connacht under the cosh.
"We are frustrated with our performance. That was a huge opportunity missed. I thought when we got within one point we would have pushed on, because we had a bit of momentum, and that is what is disappointing," conceded Elwood.
"Both teams were in a difficult place. This could kickstart their season and we have an opportunity next week to see if we can bounce back.
"I don't think Gloucester were at their best, but they did enough to win the game, and I wouldn't mind just doing enough to win a game.
"We were both desperate for the win, but I certainly believe that ourselves, as the home team, should have taken the initiative to push on and really go for the jugular, particularly when there was only one point in it."
Connacht, with several key players unavailable through injury including openside Johnny O'Connor, are also facing fresh concerns. Ray Ofisa injured his knee in the first half and is expected to miss the return fixture next weekend at Kingsholm.
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