Ed Slater, Julian Salvi and Steve Mafi scored the tries in a 27-26 win at Kingston Park, though there were anxious moments at the end when Falcons looked to line up a drop-goal attempt for fly-half Jimmy Gopperth.
“It was important for this group of players to get that first win and have that winning feeling,” said director of rugby Cockerill, coming after successive narrow defeats by Exeter and Wasps.
“Everybody needs to get that first win. It hasn’t been for the lack of effort and at times we’ve been playing some good rugby, but mistakes have cost us.
“We got ahead, then Falcons were in front and we could have let it slip, but we kept playing and kept our discipline not to give away a penalty at the end. Newcastle could have won it, but you could say we could have won in our first two games too, that is how tight it is at the moment.
“The lads are working as hard as they can and at some point you get a little luck in return and maybe that was us today.”
Tigers certainly had to dig deep, especially as the visitors lost the experience of Julian White and George Skivington by half-time, and then replacement prop Kieran Brookes was injured after less than four minutes on the pitch.
“It is not perfect, but the spirit and the will to win and the workrate is there,” said Cockerill.
“Tom Youngs came on and did a good job. His attitude is outstanding. Graham Kitchener ran things well and Michael Holford did a really good job too. He was keen to help us out and I thought he fronted up really well.”
Tigers prop Boris Stankovich, the Aviva man of the match, added: “It has been a tough couple of weeks for us so it was important to come here and get a win. It was a good hard-fought win.”
Tigers, with six points from the opening three games, host defending Aviva Premiership champions Saracens at Welford Road in round four this Saturday. Kick-off is at 2.30pm and tickets are available now.