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Mark Van Gisbergen made his first start for Wasps since returning from injury
The England U20 scrum half raced over a minute from time to add to earlier tries from Paul Sackey, Tom Rees and Tom Voyce and wrap up a 26-13 win which sets up next Sunday's visit from Newport Gwent Dragons in the third and final group game.
Wasps welcomed back Mark van Gisbergen at full back for the first time since he injured his knee on the opening day of the season, with Tom Rees wearing the armband for the first time in the absence of Raphael Ibanez.
Phil Vickery was also rested - ruled out with a late training injury – and with Pat Barnard named in the starting line up and Michael Holford on the bench, Wasps fielded what was just about their strongest team available.
Newcastle, including a number of squad players, got the first points on the board through a Rory Clegg penalty after three minutes but Wasps made the better start with a Josh Lewsey surge taking the game into Falcons' territory.
Van Gisbergen and Sackey offered support and Wasps were in a good position until ill-discipline and then a knock-on slowed their momentum.
Danny Cipriani in action for Wasps
The game was then held up for almost ten minutes as Jamie Noon went down after a collision in midfield. After receiving lengthy treatment on the pitch, he was stretched off in what appeared to be a neck brace.
When play resumed Wasps immediately went back on the front foot and they were finally rewarded after 11 minutes when Paul Sackey went over from Van Gisbergen's offload.
It was a simple score, not converted by Danny Cipriani, but there was plenty to encourage the Wasps coaching team in the build-up.
First Sackey gathered a box kick cleanly then, from Cipriani's high kick, Wasps forced a turnover to get good field position and after quick ball from the base of the ruck, good hands from Jeremy Staunton and Van Gisbergen ensured they made it count.
There were nearly in again when Tom Voyce came off his wing just afterwards, but Sackey was unable to take his close-range pass.
Wasps retained possession though, and had Newcastle scrambling when Joe Worsley attacked from the base of the scrum.
A knock on cost them that time and when Newcastle turned their second foray into Wasps territory into points, Clegg again on target with a 31st minute penalty, McGeechan's team found themselves trailing 6-5.
It was an unlikely score. Wasps were nearly playing well but each time they threatened to put something together an error or infringement ended the attack.
Wasps' Josh Lewsey
It happened again after 34 minutes, when Van Gisbergen exploded down the line. Voyce was on his shoulder to keep the move alive but Rees was just unable to take his pass, fumbling it forward.
Another chance went begging when Cipriani was off-target with his second kick of the night five minutes before half time.
From just about the same place as his conversion attempt, the England fly-half pulled his kick again, although this time only by a matter of inches.
Wasps did go the break in front though, when they were rewarded for a monumental forwards effort that battered the heart out of the Newcastle defence.
Retaining possession superbly, they strung together eight phases of play within yards of try line before captain Rees finally ploughed over near the posts.
Cipriani got off the mark with a routine conversion and Wasps went down the tunnel 12-6 in front.
They had a great chance to extend their lead just after the restart when their scrum smashed through the Newcastle pack to set up an attacking line-out.
Newcastle held on, but only for a few minutes.
After 50 minutes Ollie Phillips, back-tracking near the corner flag, dropped Cipriani's crossfield kick and Voyce had the simplest of
tasks to dot down.
From out wide Cipriani made the conversion to stretch Wasps' lead to 19-6.
Van Gisbergen, who was having a storming game, nearly picked his way over to wrap up the bonus point, and Sackey would have done with a kinder bounce, while a long-range Cipriani penalty floated just wide as the visitors continued to dictate.
When Kiwi star Carl Hayman limped off with 15 minutes left, that should have signalled the end of the Falcons' fight.
Instead though, they dug deep and penned Wasps in their own 22.
Pat Barnard was sent to the sin bin with eight minutes remaining and his bulk was sorely missed just moments later when replacement Rob Vickers burrowed over.
Clegg's conversion closed the gap to six points, and Wasps invited them back further when they won a penalty with four minutes left, but rather than taking the three points and moving two scores clear, they kicked for touch instead.
To make matters worse, they kicked the ball dead and handed over possession.
But moments later that was all forgotten as Simpson ran home from 30m to secure the bonus point that could make all the difference in Wasps' bid to reach the last four.