The Foxes forward shook off pre-match concerns over a hip injury and blasted home in first-half stoppage time to match former Manchester United striker Van Nistelrooy's streak set in 2003.
Leonardo Ulloa laid on the record-equalling effort before heading home himself, and substitute Shinji Okazaki added another as Leicester's incredible season continued with success at St James' Park.
It meant Claudio Ranieri's men moved above lunchtime leaders Manchester United, who won 2-1 at Watford, with Leicester and United meeting next weekend at the King Power Stadium where Vardy can eclipse van Nistelrooy's record against the Dutchman's old club.
Watford looked to have snatched a point when Deeney's 87th-minute spot-kick cancelled out Memphis Depay's early opener, but the Hornets captain bundled in Bastian Schweinsteiger's cross moments later to give Louis van Gaal's side a valuable three points.
The win means United are now unbeaten in eight in all competitions.
Liverpool produced a stunning performance as they beat title favourites Manchester City 4-1 at the Etihad Stadium.
On what proved a chastening night for Raheem Sterling against his old club, Liverpool enjoyed a whirlwind start, with an own goal from Eliaquim Mangala and strikes from Philippe Coutinho and Roberto Firmino inside 32 minutes.
Sergio Aguero marked his return from injury with a superb effort to reduce the arrears, but Martin Skrtel secured three points for the visitors nine minutes from time.
Liverpool's performance was some statement as they continue to build momentum under new boss Jurgen Klopp, while City were simply run ragged.
Barclays Prem
Swansea 2 - 2 B'mouth
W Brom 2 - 1 Arsenal
Man City 1 - 4 Liverpool
So'ton 0 - 1 Stoke
Newcastle 0 - 3 Leicester
Chelsea 1 - 0 Norwich
Everton 4 - 0 A Villa
Watford 1 - 2 Man Utd
Arsenal would have gone top had they won at West Brom but they blew their chance as Santi Cazorla missed a late penalty.
James Morrison and Mikel Arteta's own goal cancelled out Olivier Giroud's first-half opener as the Gunners suffered a frustrating afternoon and lost 2-1 at The Hawthorns.
Barclays Prem
1 Leicester 28
2 Man Utd 27
3 Man City 26
4 Arsenal 26
5 Tottenham 21
6 West Ham 21
7 Everton 20
8 So'ton 20
9 Liverpool 20
10 C Palace 19
11 Stoke 19
12 W Brom 17
13 Watford 16
14 Swansea 14
15 Chelsea 14
16 Norwich 12
17 Newcastle 10
18 B'mouth 9
19 S'land 6
20 A Villa 5
Cazorla slipped and sent his spot-kick over in the final 10 minutes and Mesut Ozil hit a post, but Arsenal could not recover from a nightmare five minutes just before half-time despite dominating.
Aston Villa slumped to a heavy first defeat under Remi Garde in his second game in charge as braces from Ross Barkley and Romelu Lukaku condemned the bottom club to a 4-0 thrashing at Everton.
Barkley put the Toffees ahead in the 17th minute, finishing at the far post when Brad Guzan turned Gerard Deulofeu's effort his way.
Lukaku then headed home in the 28th minute, before poor Villa defending allowed Barkley to add another three minutes before half-time.
Belgium forward Lukaku notched his second just before the hour, with Barkley and Deulofeu again involved on an afternoon where the young trio truly shone.
It will have been a hugely concerning spectacle for Garde, whose side remain five points adrift of safety at the foot of the standings.
Jose Mourinho was back in the dugout and Diego Costa returned to the scoresheet as Chelsea claimed a much-needed 1-0 win over Norwich.
Costa ended a six-match barren spell, prompting chants of 'we are staying up', as the champions secured only a fourth win in their 13th Premier League game of the season to edge away from the bottom three.
It was also 1-0 at St Mary's where Bojan Krkic fired away-day specialists Stoke to victory against Southampton.
The former Barcelona forward tucked in his third goal of the season after 10 minutes, before the miserly Potters shut up shop to register a fourth straight clean sheet on the road.
Andre Ayew came to Garry Monk's rescue as Swansea fought back from a two-goal deficit to draw 2-2 with Bournemouth.
Josh King and Dan Gosling had sent the visitors into a commanding lead after 26 minutes to give the Cherries hope of a first win in eight games.
Ayew pulled a goal back within two minutes of Gosling's strike with a classy back-heel and then won a penalty, albeit in dubious circumstances, which Jonjo Shelvey stroked home before the break.