Jagr scored during a first-period power play and Straka struck in the second period as the Rangers won for only the third time in 13 games against the Senators since the 2002-03 season.
Brendan Shanahan scored his 22nd goal of the season into an empty net and Lundqvist did the rest for New York (15-10-4), which won for the second straight game after dropping four in a row.
Jason Spezza, with a power-play marker, had the lone goal for Ottawa (15-14-1).
After winning four straight games and eight of nine, the Senators have now lost back-to-back contests.
Ray Emery stopped 26 shots for Ottawa, which slipped to 6-7-0 at home before a sold-out Scotiabank Place crowd of 19,192. New York's last victory here came March 21, 2002.
The Rangers finished 1-for-4 on the power play.
The Senators, playing without captain Daniel Alfredsson (hip) and defenceman Wade Redden (chest) because of injuries, also went 1-for-4 with the man advantage.
The Rangers opened the scoring late in the first period while the Senators were playing two men down after consecutive penalties to Jason Spezza and Antoine Vermette.
Jagr threaded a perfect cross-ice pass to Straka and his one-timer from a sharp angle beat Emery to make it 1-0 at the 17:02 mark.
The point extended Straka's scoring streak to nine games, during which he has seven goals and nine assists.
Jagr was at it again just more than six minutes into the second period when he took a pass in the neutral zone and wound his way into the slot before firing a wrist shot over Emery's shoulder to double the Rangers' advantage.
Spezza cut the lead in half late in the period, redirecting Christoph Schubert's shot from the point during an Ottawa power play.
Spezza now has at least a point in eight straight games with six goals and six assists in that stretch.
The Senators had a great chance to tie it in the final minute of the second but Lundqvist made a glove save to deny Dany Heatley on a breakaway then followed up with pair of stops on the ensuing rebounds.
Notes: The Senators recalled centre Serge Payer from AHL Binghamton on Friday to take Alfredsson's place in their lineup . . . Rangers defenceman Sandis Ozolinsh was a healthy scratch for the second straight game . . . Ottawa left immediately after the game to begin a road trip that will see the team play four games in seven days . . . The Rangers returned home and plays host to the Florida Panthers on Sunday.