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Victory moved United two points in front of Chelsea with two games to play as they once again demonstrated the ability to score important late goals, this time in a match which was all square at 1-1 with 10 minutes left.
However, Chelsea will regain top spot in the Premier League if they win at home to Stoke City at Stamford Bridge on Sunday.
Giggs's successful 58th minute penalty, awarded after Patrice Evra was brought down in the box by Benoit Assou-Ekotto, saw United - without injured England striker Wayne Rooney - go in front.
But Tottenham, fresh from wins over Arsenal and Chelsea, equalised in the 70th minute when injury-prone central defender Ledley King headed in a corner.
United, though, hit back when Portuguese ace Nani broke down the right and chipped goalkeeper Heurelho Gomes.
Veteran midfielder Giggs then sealed United's victory with his second spot-kick four minutes from time after Nani went down under a challenge from Wilson Palacios.
Tottenham will lose fourth place, and the last Champions League qualifying spot on offer to English clubs, if Manchester City win away to Arsenal later on Saturday.
