Giggs opened the scoring from the spot in the 58th minute after Spurs defender Benoit Assou-Ekotto brought down Patrice Evra.
The visitors then enjoyed their best spell of the game and captain Ledley King levelled with a header in the 70th minute.
But two goals in the last 10 minutes proved to be decisive for United. First Nani brilliantly chipped home in the 81st minute and five minutes later Giggs slotted home his second penalty of the afternoon to clinch victory and pile the pressure back on Chelsea.
At the other end of the table, West Ham midfielder Scott Parker scored a second-half wonder strike as the Hammers defeated Wigan 3-2 to ease their relegation worries.
The Hammers got off to the worst possible start when Jonathan Spector put through his own goal in the fourth minute.
But striker Ilan levelled for Gianfranco Zola's side in the 31st minute and Radoslav Kovac put the hosts 2-1 in front in first-half stoppage time.
Hugo Rodallega equalised early in the second period but Parker fired the winner to leave Hull all but relegated.
Hull went down 1-0 at home to Sunderland to leave themselves six points behind the fourth-bottom Hammers - and with an inferior goal difference - with two games remaining.
The decisive goal was scored in the seventh minute at the KC Stadium by Sunderland striker Darren Bent, with Jimmy Bullard missing a first-half penalty for the home team.
Hull's problems worsened when on-loan striker Jozy Altidore was sent off deep into first-half stoppage time for a touchline altercation with Sunderland's Alan Hutton, who was also given his marching orders.
Wolves moved a step closer to mathematically ensuring their Premier League status thanks to a 1-1 draw with Blackburn at Molineux.
A second-half equaliser from substitute Sylvan Ebanks-Blake was crucial for Mick McCarthy's side.
Wolves had trailed to Ryan Nelsen's 27th-minute strike but Ebanks-Blake pounced to earn the hosts a point.
Relegated Portsmouth produced a storming second-half fightback to draw 2-2 at Bolton.
The home side had eased into a 2-0 half-time lead over Pompey with Ivan Klasnic scoring in the 26th minute and Kevin Davies adding a second two minutes later.
But a brace from Aruna Dindane gave Pompey an unlikely share of the spoils at the Reebok Stadium.
Burnley will now be relegated if they fail to beat Liverpool at Turf Moor tomorrow.