In Group 'A', the Nepalese-Russian Giri came close to defeating the world champion with black pieces but Anand somehow managed to pose enough problems in his opponent's path, which is typical when the Indian ace is defending.
The drawn result, however, find Anand half a point behind leader Hikaru Nakamura of the United States, who also split points against former world champion Vladimir Kramnik of Russia in his 12th round encounter.
With just one round remaining in the category-20 super tournament, Nakamura tops the leaderboard on 8.5 points while Anand is close behind on eight points.
In other matches, Magnus Carlsen defeated Wang Hao of China to share the third place with Levon Aronian of Armenia on 7.5 points while Kramnik slipped to fifth spot with seven points.
In the final round, Anand will have black against Ian Nepomniachtchi of Russia while Nakamura will play with the same colour against Wang Hao.
Giri went for a Semi Slav against Anand that left the Indian in real problems.
The opening saw young Giri, 16, coming up with a Novelty early in the game and Anand with his nonchalant play, got an optical advantage with routine manoeuvres.
It was clear from the moves in the middle game that Giri understood what he was up to. Anand ran a little short of time comparatively and had to find some really fine defence after the first time control.
"I wasn't happy with my play. I just drifted, making a move according to one plan and then another in line with a different one. I had the feeling I was losing, although there was never anything concrete," Anand told reporters after the 50-move draw.
Against Kramnik, Nakamura decided against taking any risk and drew quickly that probably put a bit of pressure on Anand to catch up.
The American sounded confident about the last round against Wang Hao.
"Tomorrow, I have black against Wang Hao. At least, he's not Kramnik. So I'll have some chances in the final round," Nakamura said.
Grandmaster Surya Shekhar Ganguly finally managed a win at the expense of top seed Radoslav Wojtaszek of Poland in the 'B' group.
Both Ganguly and Wojtaszek have worked for Anand as second in the last two world championship and the victory was immensely satisfying for the Indian.
The 'B' group is headed for a tense finish as the two leaders Luke McShane of England and David Navara of Czech Republic take on each other in the final round.
India's Tania Sachdev seems poised for an improved finish in the 'C' group after a well-played victory against Mark Van Der Werf of Holland.
With 6.5 points in her kitty, Tania has defied all odds to perform way above her last seeding in the 14-players tournament.
The situation here is as tense as the 'B' group as leader Daniele Vocaturo of Italy meets 14-year-old Illya Nyzhnyk of Ukraine in the showdown for the title in the last round.
Results after round 12 Group 'A': Viswanathan Anand (IND, 8) drew with Anish Giri (NED, 6); Ruslan Ponomariov (UKR, 6) drew with Alexei Shirov (ESP, 3.5); Erwin l'Ami (NED, 4) drew with Levon Aronian (ARM, 7.5); Jan Smeets (NED, 4) lost to Alexander Grischuk (RUS, 4.5); Magnus Carlsen (NOR, 7.5) beat Wang Hao (CHN, 5.5); Hikaru Nakamura (USA, 8.5) drew with Vladimir Kramnik (RUS, 7); Maxime Vachier-Lagrave (FRA, 7) beat Ian Nepomniachtchi (RUS, 5.5).
Group 'B': Laurent Fressinet (FRA, 5) beat Li Chao (CHN, 5); Vladislav Tkachiev (FRA, 7) beat Wouter Spoelman (NED, 4); Friso Nijboer (NED, 3.5) lost to David Navara (CZE, 8); Luke McShane (ENG, 8) beat Wesley So (PHI, 7); Radoslav Wojtaszek (POL, 6) lost to SS Ganguly (IND, 5); Le Quang Liem (VIE, 7) drew with Zahar Efimenko (UKR, 7.5); Gabriel Sargissian (ARM, 7) drew with Jon Ludvig Hammer (NOR, 4).
Group 'C': Benjamin Bok (NED, 6) beat Roeland Pruijssers (NED, 4); Mark van der Werf (NED, 4) lost to Tania Sachdev (IND, 6.5); Sebastian Siebrecht (GER, 4.5) lost to Mark Bluvshtein (CAN, 6.5); Ivan Ivanisevic (SRB, 7.5) beat JW de Jong (NED, 4); Kateryna Lahno (FRA, 7) drew with Murtas Kazhgaleyev (KAZ, 6.5); Dariusz Swiercz (POL, 7) beat Daniele Vocaturo (ITA, 8.5); Illya Nyzhnyk (UKR, 8) beat Robin van Kampen (NED, 4).