Despite a poor first half from the hosts in which they struggled to ever get out of first gear, they took the lead on 28 minutes when James Bree’s corner was punched up rather than out by visiting keeper Bradley Collins, Allan Campbell prodding the loose ball over the line before Gabe Osho made sure.
Barnsley levelled just before the break through Carlton Morris, but the introduction of Fred Onyedinma 10 minutes into the second half saw the substitute immediately win a penalty after being felled by Collins, Adebayo slamming into the top corner.
The final half an hour saw Luton manage the game with very little scares, as they climbed up to seventh in the table, with a sixth win from nine in the second tier, as Jones said: “It was a classic win, I’m happy with that, it wasn’t one for the purist, but sometimes you've got to dig them out, especially at this level.
“There's been teams that do fantastically well year in year out in the Championship, that learn how to just win games and that’s what we’re doing at the minute.
“They're very difficult to play against, they pack the midfield, they've got a lot of young, technical players that can cause you problems, but we needed to cause them more problems going that way, show more composure than we should have.
“We mess about before half time, try a backheel when it’s not on, and let them back into the game.
“We should have had a penalty, so we should have been two up at half time, and if we're just professional in what we do, then we see that game out.
“We haven't played the best, but we see that game out.
“Once we made the tactical change (bringing Fred Onyedinma on) I thought we were much, much better.
“They went gung ho late on, and I thought we saw the game out really well.
"So even though it wasn't a classic, I thought it was a wonderful, wonderful win.”