Sen. Bernie Sanders won Indiana’s primary Tuesday, giving his flagging presidential campaign ammunition to press forward and making plain front-runner Hillary Clinton’s persistent trouble unifying the Democratic Party.

With 80% of precincts reporting, Mr. Sanders had 53% to Mrs. Clinton’s 47%, according to the Associated Press.

Even with the win, Mr. Sanders still faces a wide deficit in convention delegates and virtually no chance to close the gap. He picked up at least 43 of the 83 pledged delegates at stake, but Mrs. Clinton took at least 37, according to an AP tally.

Still, the late-in-the-game win by the Vermont senator creates an odd split-screen for the Democrats, with the Clinton campaign increasingly focused on Republican Donald Trump even as Mr. Sanders lays bare her weakness with a large slice of Democrats. Exit polls showed he again beat her among young people, white voters, political independents and people who most value honesty in a candidate.

The senator’s win in Indiana may make it more awkward for Mrs. Clinton to look past him, as she did this week. Confident in her delegate lead, she spent much of Tuesday campaigning in the fall battleground state of Ohio. Her campaign in recent days began hiring staff across states expected to be competitive this fall, and her advisers were in the early stages of a vice-presidential selection process.