Jason Tovey did get the visitors on the scoreboard with a three-pointer before the interval, and after the break tries for Sarel Pretorius and James Thomas brought Newport to within two points of the home side in the closing stages.
However, Danie Poolman scored a breakaway try for Connacht and Newport had to settle for a bonus point thanks to Tom Prydie's long-range penalty at the death.
"We were very nervous in the first 20 minutes playing into a stiff breeze and we conceded nine easy points. It wasn't good play from Connacht really, it was bad play by us," said Jones.
"As the game went on we grew and got better and better. We got to 22-20 with five minutes on the clock and were confident the momentum was with us but they ran the length and that was typical of what went on in the night.
"We are not disappointed, the performance was good and it bodes well for the rest of the year.
"We really looked positive in the second half and to see Tom Prydie put a 45-metre kick over for a bonus point was a just reward.
"Players are very disappointed that we didn't come away with a victory but as a coach you've got to look further into the result and performance. It bodes well.
"When you come to Galway it's a really difficult game and if you commit unforced errors, as we did in the first half to gift them 14 points, it's too much to claw back."
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