Gateshead's forward resources were stretched before this match and they lost Ben Sammut, to a broken arm, and Andrew Hick, to a torn calf, in the first few minutes. Unsurprisingly, they were slow to settle, and almost conceded a try to Rivett's clever chip and chase before Daryl Powell punched a hole in their defence to send Francis Cummins away.
The Thunder hit back for the first time when Willie Peters sent Danny McAllister through a huge gap. Ryan Sheridan sneaked in to put Leeds back in front, but Peters' full array of scrum-half skills again paid off for Gateshead when he planted a perfect kick into the corner for Matt Daylight.
At that stage, the only difference between the sides was Iestyn Harris's extra conversion, but it all went wrong for the Thunder in the eight minutes before half-time. There seemed no particular danger when Harris's pass found Rivett, but he somehow managed to twist his way over the try line.
Back under pressure when Ian Herron kicked dead from the restart, Gateshead were split by Jamie Mathiou's pass to Harris. Although Deon Bird stopped him, it was at the expense of a penalty and a sin-binning and Paul Sterling soon found an unguarded blind side to claim a fourth Leeds try.
After the break, Harris added a penalty and then converted after Sheridan's burst through the middle had set up the substitute, Marcus St Hilaire.
Gateshead showed their resilience when Will Robinson took a return pass from Craig Simon to score their third try, before the alert Rivett's second three minutes from time.
The Gateshead coach, Shaun McRae, said the club would step up its search for a new hooker and complained bitterly about the incident in which Sammut was injured. "I'm astounded that it was not put on report," he said.
Leeds Rhinos: Harris; Rivett, Golden, Cummins, Sterling; Powell, Sheridan; McDermott, Newton, Farrell, Morley, Hay, Glanville. Substitutes used: St Hilaire, Jackson, Mathiou, Sinfield.
Gateshead Thunder: Bird; Herron, Grogan, Simon, Daylight; Robinson, Peters; Lee, Sammut, Hick, Felsch, Maher, Grimaldi. Substitutes used: Wilson, Maiden, McAllister, S Allwood.
Referee: R Smith (Castleford).
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