The Falcons asserted themselves strongly in the opening minutes of the match as Matthew Tait was forced to concede a line-out close to his tryline.
The home side collected the line-out and used the power of their pack to allow Scott Lawson to put them ahead with a short run to the line.
The visitors got into the match when they won a penalty from a scrum, which allowed Freddie Burns to fire over from close range and bring the Tigers within four points.
Burns then showed good strength with his leg to kick from distance as the home side were penalised at the breakdown to make it a one-point game.
Burns continued his fine display in the second half as he fired over from close range to bring Leicester within three points.
The Tigers increased the pressure on the Newcastle line towards the end of the match and they got their reward when Bell touched down after the home side had failed to gather a high ball. Burns then added the crucial conversion to put his side out of range of a penalty or drop-goal.
The Falcons tried to hit back, but a handling error presented the Tigers with the ball, allowing them to see out the clock for the victory.