The famous “Hail Mary” pass was Dallas Cowboys quarterback Roger Staubach’s throw to wide receiver Drew Pearson in a December 28, 1975 football game against the Minnesota Vikings. The Cowboys had been losing 14-10, but that last-second “Hail Mary” pass play helped them to a 17-14 victory. A football “Hail Mary” play is any last-second, long-shot, low-odds play, a “prayer” that is sometimes answered.
 
It’s not generally known, but Roger Staubach was quoted using the “Hail Mary” football terminology in 1972 and 1974. Also, Notre Dame has a “Hail Mary” tradition that began in a 1922 football game against Georgia Tech.
 
 
Photo Gallery of Great Moments
NFC Divisional Playoff Game, Cowboys versus Vikings, December 28, 1975,
Metropolitan Stadium, Bloomington, MN
With less than two minutes to play and trailing 14-10 the Cowboys took over possession of the ball on their own 9-yard line.  QB Roger Staubach led the Cowboys attack by hitting WR Drew Pearson on five different passes including a clutch 22-yard forth down conversion at the 50-yard line.  On second down, and only 36 seconds remaining, Staubach threw a desperation pass to WR Pearson who was being covered by CB Nate Wright.  The pass was a bit under thrown and there was some contact between Pearson and Wright.  Wright fell to the ground, Pearson bobbled the ball on the five-yard line.  He eventually was able to pin the ball against his right hip to gain possession of the ball.  He then strolled into the endzone for the winning touchdown.  QB Roger Staubach, in a post-game interview said “It was a Hail Mary pass.”  A moniker that would forever stick describing this particular catch and created a new lexicon for all similar pass plays to follow.  Sadly, Vikings QB Fran Tarkenton’s father died of a heart attack while watching the game.  His name was Dallas..       
 
Wikipedia: Hail Mary pass
A Hail Mary pass or Hail Mary play in American football is a forward pass made in desperation, with only a very small chance of success. The typical Hail Mary is a very long forward heave thrown at or near the end of a half where there is no realistic possibility for any other play to work, though the most famous were thrown at the end of a game. The phrase derives from the name of a prominent Roman Catholic prayer to the Virgin Mary. The success of this play is unlikely due to the general inaccuracy of the pass.
   
(Oxford English Dictionary)
Hail Mary, phr. and n.
Chiefly N. Amer. Sport. A desperate play or manoeuvre having a very low likelihood of success; (Amer. Football) a long pass thrown into or near the end zone by a losing team as time is running out. Chiefly attrib.
1972 Sporting News 15 Jan. 8/2, I asked if he really saw Pat or if he was throwing it away. He [sc. Roger Staubach] said, ‘Let’s just call it my Hail Mary Play.’
         
Newspapers.com (An identical January 4th Associated Press story.—ed.)
10 January 1932, Portsmouth (OH) Times, pg. 12, col. 2:
“Hail Mary”
Play Wins
Football Game
   
NEW YORK, Jan. 9—Gus Welch retained the “Brown Derby” at the annual banquet of the American Football Coaches’ association, but Jim (Sleepy) Crowley, one of the Four Horsemen of Notre Dame and now a coach at Michigan State, brought down the hall with this one:
   
“In 1922 Notre Dame had nine sophomores on the team that went to Atlanta to play Georgia Tech,” Crowley related. “In the first half Tech got a field goal and things looked pretty dark for us. In the third period Layden punted to Red Barron, who muffed. We recovered on the 20-yard line and tried three plays in vain. It was fourth down.
   
“It so happened that we had a Presbyterian on the team. He stopped play and said to us, ‘Boys, let’s have a Hail Mary.’ Well, we prayed, and Layden soon went over for a touchdown.
   
“Believe it or not, the formula was repeated. Again Layden kicked, again Barron fumbled, again we tried three plays in vain. ‘Let’s have another Hail Mary,’ said the Presbyterian. Well, again Layden went over for a touchdown.
   
“After the game I discussed the strange series of events with our Presbyterian. ‘Say, that Hail Mary is the best play we’ve got,’ he exclaimed.”
 
20 December 1939, Charleston (WV) Gazette, pg. 18, col. 4:
Crowley related humorous slants on…the 1922 game with Georgia Tech won by Notre Dame 14-3 on what the Presbyterian quarterback said was due to his “hail Mary” play.
 
20 December 1965, Ironwood (MI) Daily Globe, pg. 10, col. 4:
Tom Flynn called it his “Hail Mary shot” and here’s how he described it: “I just grabbed it and threw it,” said the Marquette captain whose shot in the final two seconds gave Marquette a 75-74 basketball victory over Washington Saturday night and the Milwaukee Classic championship.
 
15 January 1972, Sporting News, pg. 8, col. 2:
Hardin Recalls Feat
His former Navy coach, Wayne Hardin, recalled recently: “I remember a game at Michigan. We were on the 20 and Roger rolled right and got hammered in. He was bobbing and weaving and kept retreating, back to the 30. He was surrounded and upended.
 
“He was parallel to the ground, his feet in mid-air and he threw a pass to Pat Donnelly, who made a one-yard gain.  Afterward, I asked if he really saw Pat or if he was throwing it away.
 
“He said, ‘Let’s just call it my Hail Mary Play.’”
 
So the Cowboys have got Staubach and his “Hail Mary” plays, they’ve got running backs like Duane Thomas and Calvin Hill, they’ve got top receivers, a talented offensive line and the Doomsday Defense.
 
29 August 1974, Gettysburg (PA) Compiler,  pg. 14, col. 7:
Everyone is back from the team that won the National Conference West before losing at Dallas, 27-16, in the playoffs on what Cowboys’ quarterback Roger Staubach calls “a couple of Hail Mary pass plays.”