Baker made his name as the robot in the first Star Wars film in 1977 alongside Anthony Daniels' C-3PO character.
Lucas said Baker was "a real gentleman" and Hamill, who plays Luke Skywalker, said he had lost "a lifelong friend".
Born in Birmingham, Baker's other films include Time Bandits and Flash Gordon.
Star Wars R2-D2 actor Kenny Baker dies
Preston pays tribute to R2-D2 actor
Lucas said: "Kenny Baker was a real gentleman as well as an incredible trooper who always worked hard under difficult circumstances.
"A talented vaudevillian who could always make everybody laugh, Kenny was truly the heart and soul of R2-D2 and will be missed by all his fans and everyone who knew him."
Writing on Twitter, Hamill said: "Goodbye #KennyBaker A lifelong loyal friend-I loved his optimism & determination He WAS the droid I was looking for!"
Anthony Daniels, who played C-3PO, added his voice to the tributes, tweeting: "Sad to hear of Kenny's passing. One of the truly original cast, so famed for his iconic role as R2. He'll always be remembered by so many fans."
C-3PO may have once described him as a "nearsighted scrap pile" and an "overweight glob of grease", but he like film fans all over the world has always had a huge degree affection for R2-D2.
Artoo's appearance may have been likened by some to that of an industrial vacuum cleaner, but Kenny Baker worked hard to make sure he managed to convey the right emotions at key moments. Kenny used an array of movements ranging from subtly shuffling himself forward to rocking from side to side to give the robot personality.
Together with the Ben Burtt-designed whistles and beeps that are Artoo's voice, it made the droid as real a character as any other in the saga - human or alien.
We were all also able to relate to Artoo; here was a character who bickered with his friend C-3PO, occasionally manipulated those around him and displayed his courage when necessary.
We may never have been able to understand exactly what the astromech droid was saying, but we always seemed to know what he was thinking, thanks to the way Kenny Baker was able to humanise him.
Game of Thrones star Gwendoline Christie, who plays Captain Phasma in Star Wars: The Force Awakens and the upcoming Star Wars: Episode VIII, tweeted: "Rest in peace Kenny Baker. It was a pleasure to meet you & thank you for all the joy you brought us."
Chris Weitz, one of the screenwriters for Rogue One: A Star Wars Story, which is due out this December, wrote the touching tribute: "RIP Kenny Baker. You made my childhood better, " while Greg Grunberg, who played Snap Wexley in Star Wars: The Force Awakens, simply added: "RIP 'Star Wars' Kenny Baker. So sad."
Ant Man writer and Shaun of the Dead director Edgar Wright paid tribute by tweeting: "RIP R2D2. Farewell Fidget, Time Bandit. Goodbye Mister Kenny Baker."
Game of Thrones star Daniel Portman, who plays squire Podrick Payne, added: "Met Kenny Baker multiple times, a real character and a man who will live on forever as one of Cinema's great characters. Rest in Peace. X"
Star Wars set decorator Roger Christian, who created the first R2-D2, told BBC Radio 5 live about how Baker was very nearly not in the film.
The 3ft 8in (1.12m) actor was in a joint theatrical act called the Mini Tones with friend Jack Purvis, and the duo had just made the final of talent show Opportunity Knocks, so Baker left the Star Wars team.
"There was minor panic because there wasn't anybody else who could make the scale right and if we didn't have R2-D2 George didn't have a film, Star Wars wouldn't have been made," Mr Christian said.
Baker returned after the team agreed to employ Purvis as well.
Of the R2-D2 costume, Mr Christian said: "Kenny was always complaining because it was difficult, it was so tight in there, we had to put rubber round him, foam rubber, and he couldn't move it, he could kind of shake it a bit, but that was about it...
"I found a fighter pilot's harness and we fitted that inside R2-D2 so Kenny could wear R2-D2 like a rucksack.
"And he actually stumbled forward about three steps and crashed over - we could hear him inside saying 'let me out!' But that was it."
He added: "He would endure it for hours and hours he was a good sportâ€¦ he would make jokes all the time."
Baker's nephew, Drew Myerscough, said he had cared for Baker for "eight or nine years" after he developed respiratory problems.
He said his uncle, who lived in Preston, had a passion for wildlife documentaries and had "a liking for lasagne".
"He was just a normal, down-to-earth, regular guy that enjoyed life," he told the BBC, adding they "rarely" discussed Star Wars, adding: "His fans worldwide kept him going and he loved nothing more than going to conventions and meeting everybody - it really gave him that extra lease of life."
Other tributes came from Ewan McGregor, who played Obi-Wan Kenobi in the three Star Wars prequels, who tweeted: "So sorry to hear about this. It was lovely working with Kenny."
Actor Daniel Logan, who played Boba Fett in 2002's Star Wars: Episode II - Attack of the Clones, added: "So sad to hear one of my dearest friends passed away. Rest in peace Kenny Baker. Love you. Will miss you!"
David Prowse, who played Darth Vader in the original Star Wars trilogy, tweeted: "Very sorry to hear about Kenny. He was a great personal friend that I thoroughly enjoyed working with. #KennyBaker"
Actor Warwick Davis, who played characters in three Star Wars films, tweeted: "#RIP #KennyBaker. Sad to say goodbye to a small man with a huge heart & personality. He paved the way for short actors of a generation. x"
Lucasfilm president Kathleen Kennedy said: "There is no Star Wars without R2-D2, and Kenny defined who R2-D2 was and is."
Mexican film director Guillermo del Toro, who directed Hellboy and Pan's Labyrinth, simply tweeted: "R.I.P Kenny Baker," while Clerks director Kevin Smith said: "Rest in Peace @starwars legend #KennyBaker - whose name I learned at age 7 because he brought #R2D2 to life. Feels like losing family..."
The studio behind the original Star Wars films, 20th Century Fox, posted a still of R2-D2 and C-3PO and said: "Rest in peace, Kenny Baker, the heart and soul of R2-D2."
Actress Morgan Fairchild, who appeared in the 1987 film Sleeping Beauty with Baker, tweeted: "Hate to hear this -lovely man-lovely soul."
After starring in the original Star Wars, Baker he went on to appear in the sequels, The Empire Strikes Back and Return of the Jedi, and the three prequels between 1999 and 2005.
He was a consultant on last year's Star Wars: The Force Awakens but British actor Jimmy Vee was already lined up to take on the role of R2-D2 in the next film, due for release in 2017.
Baker was a father of two, and his wife Eileen died in 1993.
Star Wars' George Lucas and Mark Hamill are among the big names paying tribute to actor Kenny Baker, who has died aged 81, describing him as "the heart and soul of R2-D2".