Buddy Robinson also scored his first goal for Ottawa.
Ondrej Pavelec stopped 25 shots as Winnipeg's exhibition record moved to 2-3. Ottawa is also 2-3.
Slater scored at 9:17 of the first period after he got the puck behind Ottawa's net and was barely slowed down after a check by Sens defenceman Alex Grant.
Slater then curled around into the face-off circle and fired the puck high past goalie Robin Lehner to make it 1-0.
Craig Anderson took over in net at the start of the second period after Lehner faced 13 shots.
Anderson finished by turning aside 10 shots.
Ehlers, a native of Denmark who was Winnipeg's top pick in this year's NHL draft (ninth overall), is one of the young prospects the Jets have been keeping a close eye on during training camp.
His first pre-season goal came after a quick pass from Matt Halischuk, who sent the puck across the front of the net to Ehlers, who had a wide-open side to pop in the puck at 6:22.
Ottawa featured a young lineup including forwards Curtis Lazar and Mark Stone, who was playing in his hometown for the first time as a pro.
Veterans Bobby Ryan, Milan Michalek and David Legwand were among those left at home.
The Jets sat veterans such as captain Andrew Ladd, Dustin Byfuglien and Bryan Little, but did run their expected top line of centre Mark Scheifele with Evander Kane and Blake Wheeler.
Robinson got the Sens on the scoreboard with 5:18 left in the second period on a breakaway that started with a rebound off Anderson's pads.
Senators defenceman Aaron Johnson got the rebound and then sent a long pass up the ice to a rushing Robinson, who went in alone on Pavelec and beat him with a forehand deke.
Scheifele was a target in the second period, first taking a big hit from Grant and then getting smashed from the side and onto the ice by forward Chris Neil, who was sent to the box for charging with 16 seconds left in the period.
The bad feelings continued in the third period, when Neil and Jets winger Anthony Peluso threw some punches at each other.
Winnipeg travels to Calgary for a Thursday game, while Ottawa hosts Montreal Friday.