The three-day drills, announced the day after Tsai returned from the United States, had been widely expected after China condemned her Wednesday meeting with Speaker Kevin McCarthy in Los Angeles.
China views democratically governed Taiwan as its own territory and has never renounced the use of force to bring the island under its control. Taiwan's government strongly objects to China's claims.
Beijing's announcement on Saturday also came just hours after China hosted a visit by senior European leaders.
The People's Liberation Army said it had started the combat readiness patrols and "Joint Sword" exercises around Taiwan, having said earlier it would be holding them in the Taiwan Strait and to the north, south and east of Taiwan "as planned".
"This is a serious warning to the Taiwan independence separatist forces and external forces' collusion and provocation, and it is a necessary action to defend national sovereignty and territorial integrity," the Chinese army's Eastern Theatre Command said in a statement.
Taiwan's Defence Ministry said as of 4pm on Saturday it spotted 71 Chinese aircraft, including fighter jets and bombers, crossing the median line that normally serves as an unofficial barrier between the two sides, as well as nine Chinese ships.
China was using Tsai's US visit "as an excuse to carry out military exercises, which has seriously damaged regional peace, stability and security", the ministry said in a statement.
"The military will respond with a calm, rational and serious attitude, and will stand guard and monitor in accordance with the principles of 'not escalating nor disputes' to defend national sovereignty and national security."
Chinese state media released what it said was footage of the drills, set to stirring martial music and showing warships at sea and fighter jets doing airborne refuelling. Reuters could not authenticate when or where the material was shot.
China had threatened unspecified retaliation if the meeting with McCarthy - second in line to succeed the US president, after the vice president - were to take place. Beijing staged war games around Taiwan, including live-fire missile launches, in August after then-House Speaker Nancy Pelosi visited Taipei.
A senior Taiwan official familiar with security planning in the region told Reuters the aircraft involved in the morning missions had only crossed the median line briefly.
Reuters reporters in a seaside area near Fuzhou, opposite the Taiwan-controlled Matsu islands, saw a Chinese warship firing shells onto a drill area on China's coast, part of drills announced by China late on Friday.
Writing on her Facebook page, Tsai said she had received a briefing about the security situation and that the military was at its post around the clock.
"Taiwan will stand with all democracy-loving partners in the world and jointly assume the responsibility of ensuring regional stability and prosperity," she added.
Tsai has repeatedly offered talks with China but has been rebuffed as the government views her as a separatist. She says only Taiwan's people can decide their future.
China's announcement of the military drills came hours after French President Emmanuel Macron left China, where he met President Xi Jinping and other senior leaders. Macron urged Beijing to talk sense to Russia over the war in Ukraine.
European Union chief Ursula von der Leyen, also in China this week to meet Xi, said stability in the Taiwan Strait was of paramount importance.
Xi responded by saying that expecting China to compromise on Taiwan was "wishful thinking", according to China's official reading of the meeting.
