1 Joffrey was Poisoned
Joffrey was poisoned during his own wedding to Margaery. He asked Tyrion to serve him a drink. The drink turned out to be poisoned. See Summary of Episode 2 season 4 - The Lion and the Rose.
2 Tyrion wasn't a part of it
Tyrion didn't knew the drink he was serving to Joffrey was already poisoned. See why Cersei hates Tyrion.
3 Baelish arranged it with Lady Olena
Baelish wanted to kill Joffrey for his own reasons while Lady Olena didn't want her daughter Margaery to marry such a monster , the word Olena used to describe Joffrey. See why the Lannisters hate the Starks.
4 The Poison was attached to Sansa's necklace
Dontos , the man Sansa spared his life before, Gave her a necklace as an act of gratitude. The Necklace was given to Dontos by Baelish and it had the poison in one of it's crystals in a solid form.
5 Dontos was working for Baelish
Dontos was following Baelish's orders and he knew he is being a part of a plot. This is why Baelish killed him after the mission was done.
6 Lady olenna placed the poison in the wine
During the wedding Lady Olenna took the poison from Sansa's necklace , without the later knowing, then dropped it into Joffrey's wine. See also why Eddard Stark Died.
7 Sansa had nothing to do with it
Sansa wasn't a part of the plot nor she knew what was going on. She really believed that the Neckalce was a gift from Dontos because she saved his life. See Summary of Episode 1 - Season 4 - Two Swords.