Jones' side looked set to suffer a very heavy defeat on their top flight bow when two Thomas Waldrom tries helped Leicester to a 17-0 lead.
However, the hosts showed great courage, and no little skill, to fightback through tries from Tom Arscott and Hudson Tonga'uhia and they trailed by just seven at the break.
But further scores from Geoff Parling, Julian Salvi and Vereniki Goneva sealed Leicester's victory, with fly-half Toby Flood converting all five tries and a penalty for a 13-point haul.
Championship winners Welsh needed to go through an RFU appeals process just to secure their place in the top flight, having originally been deemed not to meet the required off-field standards.
By the time that decision was reversed at the end of June, the Exiles faced a race against time to recruit a squad capable of standing up to the rigours of Premiership rugby.
And former Ospreys boss Jones admitted bringing in players so late had led to a lack of cohesion as Welsh were introduced to the harsh realities of life at the top.
He said: "What happened in the summer has been well documented, but the knock-on effect is in the detail of your game.
"We knew we just did not have enough time to put the detail into the game. We had players knocking into each other in areas of the field where they weren't sure what they should be doing and that stood out and it cost us.
"I am not saying we were good enough to win but if we had upped our game at the line-out maybe we could have challenged for a bonus point, but we failed when we had opportunities to launch attacks."
But Jones was impressed by the efforts put in by his side against the nine-time champions, and believes there is plenty to build on before their trip to title holders Harlequins on Friday.
"There is only so much tackling you can do, Leicester came in wave after wave and I was proud of the boys, they stuck in there, they gave 100 per cent and you can't ask more than that.
"There was a lot of good in our game and we now have foundations to build on and it's important we develop to become a tough team to beat. But it might take four to six weeks until we have something we are all comfortable with."
Leicester never really had to get out of second gear to record their bonus-point win, but rugby director Richard Cockerill was pleased with their start.
He said: "We come away with maximum points and it was a good workout for us, we did some good things in the first 25 minutes and some poor things in the last 15 minutes of the first half. In the second half we were much more accurate.
"But credit to Welsh they were up for the battle and got themselves back in it and if you are not on your game they will punish you.
"We did not know what to expect from them but there is enough experience in that side and physically they were very good and made life difficult for us."
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