England full-back Armitage capped a fine game by scoring a try, while debutant centre Hape's break in the final quarter set up the try which killed off Newcastle's brave challenge.
And Dan Bowden played his part, scoring 13 second-half points after coming off the bench for fly-half Steven Shingler, while Alan Thompstone's second try ensured Irish earned the bonus point.
Both teams made it clear from the outset that caution would play little or no role in their approach as attack and counter-attack saw ball repeatedly moved through hands.
And it was Newcastle who had the edge during the early exchanges, pinning the hosts back and seeing Rikki Sheriff's burst for the line foiled when he knocked on in the tackle when over the line.
The visitors took a deserved lead on 15 minutes when sustained phases inside the Irish 22 sucked in their cover before centre Jamie Helleur was released to sprint over wide on the right. Jeremy Manning's conversion attempt flew inches wide.
Irish responded within four minutes, driving the ball into the visitors' area where Jonathan Joseph's clever reverse pass armed Thompstone, who powered over the line. Steve Shingler's conversion edged them a point in front.
Back came Newcastle, whose pressure squeezed a 23rd minute penalty out of an anxious defence and Manning kicked the penalty.
Referee Dean Richards adjudged the visitors offside just outside their 22 after 26 minutes and Shingler's penalty flew high between the posts to give Irish the lead for the first time.
And it got worse for Newcastle five minutes later when they were penalised for not releasing in the tackle, presenting Shingler with a bread and butter shot in front of the posts
Irish finished the half on top and Shingler increased their lead with a penalty for a ruck offence on 40 minutes.
Manning struck a post with a drop goal attempt seconds into the second half, then booted a penalty into touch as Newcastle forced the Exiles to defend in depth just feet from their line.
They were rewarded on 49 minutes when a succession of phases created a two-on-one wide on the right and Greg Goosen chose to go for it, scoring close to the post. Manning added the conversion.
He was equally accurate with a penalty in the 53rd minute, recapturing the lead - only for Ashley Wells to concede a soft penalty with a deliberate knock-on to disrupt a pass. Dan Bowden, on for Shingler at half-time, despatched the penalty.
That raised the Irish tempo and they scored their second try three minutes later.
Jamie Gibson's break sparked the pressure and when Bowden fired a mispass wide right, Armitage was clear to do swan-dive over by the flag. Bowden missed the conversion.
Irish increased their lead on 61 minutes when Bowden boomed home a drop goal from behind a ruck, then Hape marked a solid debut with a powerful break into the heart of the Newcastle defence, breaking tackles before providing a perfect offload which armed Bowden to score between the posts and convert his own try.
Thompstone then reacted fastest to Bowden's kick-through to score his second try of the tie five minutes from time.
Joe Graham's try for Newcastle on 80 minutes was merely a consolation.
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