The match ignited with three late first-half goals, the first coming in the 35th minute as Jack Parkinson curled in superbly off the post for the visitors from Mitch Pinnock's corner.
Gateshead broke away to equalise straight from kick-off when Wes York was on hand to tap in after Jones' point-blank shot was saved.
Jones then collected Paddy McLaughlin's cross superbly to put the hosts in front in the 44th minute.
McLaughlin made it 3-1 three minutes into the second half, but Ricky Miller halved the deficit after 57 minutes with his 10th goal of the season.
Sam Jones fired his second past Mitch Walker two minutes later to restore the two-goal lead but missed out on a hat-trick when he hit the crossbar with a penalty.
Report supplied by the Press Association.
Match ends, Gateshead 4, Dover Athletic 2.
Second Half ends, Gateshead 4, Dover Athletic 2.
Substitution, Dover Athletic. Sammy Moore replaces Chris Kinnear.
Mitch Brundle (Gateshead) is shown the yellow card for a bad foul.
Substitution, Gateshead. Gus Mafuta replaces Nyal Bell.
Substitution, Gateshead. Reece Styche replaces Wes York.
Wes York (Gateshead) is shown the yellow card for a bad foul.
Aswad Thomas (Dover Athletic) is shown the yellow card for a bad foul.
Ricky Miller (Dover Athletic) is shown the yellow card for a bad foul.
Goal!  Gateshead 4, Dover Athletic 2. Sam Jones (Gateshead).
Goal!  Gateshead 3, Dover Athletic 2. Ricky Miller (Dover Athletic).
Goal!  Gateshead 3, Dover Athletic 1. Patrick McLaughlin (Gateshead).
Second Half begins Gateshead 2, Dover Athletic 1.
First Half ends, Gateshead 2, Dover Athletic 1.
Goal!  Gateshead 2, Dover Athletic 1. Sam Jones (Gateshead).
Goal!  Gateshead 1, Dover Athletic 1. Wes York (Gateshead).
Goal!  Gateshead 0, Dover Athletic 1. Jack Parkinson (Dover Athletic).
First Half begins.
Lineups are announced and players are warming up.

Last year Lincolnshire Fire and Rescue Service attended 37 incidents, compared with 16 in 2014 /2015.
The service is now carrying out special training sessions using a 28 stone (179kg) dummy.
Figures obtained by the BBC show a rise in "bariatric rescues" from four assistance requests in 2011/12.
More on this and other Lincolnshire stories.
The type of rescues logged by the fire service included removing bedroom doors and banisters, to using lifting equipment to help patients from their homes into ambulances.
Spencer Creek, technical response manger for the service said that more than 60 firefighters had taken the specialist training course.
"What we are looking at now is how the guys practically deal with the weight of the casualty," he said.
"And how do we deal with it in a number of different scenarios."
As well as the training, the service has invested in specialist equipment including, inflatable chairs and strengthened stretchers at the cost of Â£3,000 per fire engine.
East Midlands Ambulance Service has equipped all its ambulances with stretchers capable of holding patients weighing up to 30 stones (190kg).

Seamer Porter dismissed Richard Levi in the third over of the final morning.
Ryan ten Doeschate (4-31) then took 3-17 from nine overs as Adam Rossington and Azharullah fell lbw and David Murphy was caught at mid-on.
Essex wrapped up victory after lunch as Graham Napier bowled Ben Sanderson as the visitors were all out for 230.
Rory Kleinveldt finished unbeaten on 35, having hit five boundaries, and Richard Gleeson made a career-best 13 but Northants folded against the hosts' pace bowlers.
Porter, who has now claimed 22 wickets at 17.18 from his first three Championship matches of 2016, also picked up a career best of 5-46 in the first innings to help Essex maintain their unbeaten start to the season.
BBC Radio Northampton's Alex Winter: "A pretty poor all-round performance from Northamptonshire here. They bowled waywardly on the first day and have been chasing the game since.
"The batting was marginally better in the second innings but nowhere near good enough to get back in the game.
"Essex have put in an efficient display for a huge win and their second victory of the season. Promotion challenge on?"
Sam Jones hit a brace as Gateshead beat Dover at the International Stadium.
[NEXT_CONCEPT]
Firefighters in Lincolnshire have seen a doubling in the number of calls to assist the ambulance service treat obese people in one year.
[NEXT_CONCEPT]
Jamie Porter's career-best first-class match figures of 8-91 helped Essex ease to an innings and 92-run victory over Northamptonshire at Chelmsford.