TL;DR

Sinatraのjson_encoderを入れ替える
:json_encoder => some_encoder で渡せる
oj は Sinatra内部で呼ばれるmethodがないので別途作成して食わせる

ref

http://www.sinatrarb.com/contrib/json.html

sample

Gemfile

source "https://rubygems.org"

# utilities
gem "rake", '~> 10'
gem "yajl-ruby"
gem "oj"
gem "oj_mimic_json"

# rack-app
gem "thin"
gem "sinatra"
gem "sinatra-contrib"
gem "unicorn"

app.rb

require 'sinatra'
require 'sinatra/json'
require 'yajl'
require 'oj'
require 'oj_mimic_json'
require './oj_encoder.rb'

get '/oj' do
  hash = {}
  hash['key'] = 'hoge'
  hash['key2'] = 'fuga'
  hash['uuid'] = SecureRandom.uuid
  json hash, :json_encoder => OjEncoder.new
end

get '/yajl' do
  hash = {}
  hash[:key] = 'hoge'
  hash[:key2] = 'fuga'
  hash[:uuid] = SecureRandom.uuid
  json hash, :json_encoder => Yajl::Encoder
end

get '/' do
  hash = {}
  hash[:key] = 'hoge'
  hash[:key2] = 'fuga'
  hash[:uuid] = SecureRandom.uuid
  json hash
end

config.ru

require 'rubygems'
require 'sinatra'

require './app.rb'

run Sinatra::Application

oj_encoder.rb

require 'oj'

class OjEncoder

  def initialize
    ::Oj.default_options = {:mode => :compat }
  end

  def encode(value)
    ::Oj.dump(value)
  end

end