Strasburg had an MRI on the elbow earlier Thursday, one day after leaving a start in the third inning.
Lessard said the injury is not necessarily season-ending but offered no timetable for Strasburg's return.
"I'm thinking let's take care of the swelling and get him to a pain-free range of motion, and then we'll start a strengthening program," Lessard said. "I'm just worried about the next four or five days right now."
Strasburg felt a pinch in the back of his elbow Wednesday night, and was forced to make an early exit in his first start since returning from the disabled list.
"Unfortunately, it just happened on one pitch," Lessard said. "It's just an acute injury vs. an overuse injury."
Strasburg previously had been out since Aug. 17 because of soreness in his right elbow. He left Wednesday's start after 2 1/3 innings and 42 pitches, allowing a run on two hits, before NL East-leading Washington went on to defeat Atlanta 5-4 in 11 innings.
Before going on the disabled list, Strasburg allowed 19 runs over his final three starts, including nine over 1 2/3 innings in a loss to Colorado his last time out. In his first 17 starts, he was 13-0 with a 2.51 ERA.
It was Strasburg's second stint on the disabled list this season. The right-hander missed 17 games in June with an upper back strain.
After he exited Wednesday, fears were that Strasburg would need a second Tommy John surgery after having the procedure in 2010.
"It's good to hear it wasn't what everybody was speculating," manager Dusty Baker said Thursday.
"Waiting on news is always nerve-racking. You should have been with me waiting on news that I had cancer ... toughest part of my cancer was the three-month checkup, six-month checkup. Waiting on news is not very pleasant."
Strasburg signed a seven-year extension worth $175 million in May. He had an MRI before inking that deal, and Lessard said Thursday that Strasburg's elbow looks "basically the same" as it did then.
The Nationals entered Thursday night with an 8½-game lead over the Mets in the NL East.
Asked who will take Strasburg's spot in the rotation, Baker said the team hadn't decided yet.
"We have the personnel that we have," he said. "You can't count on somebody else coming in here. So we have to the best for the situation with what we have and then make a determination on who is to take his place now and possibly later or is it a better option to not have Stephen or do we not count Stephen out? Those are our options, and the Lord knows and can heal Stephen with the help of our trainers and doctors."
Information from ESPN's Eddie Matz and The Associated Press was used in this report.