Boro were held at arm's length by Dean Holdsworth's Shots for 83 minutes at the Riverside Stadium, but Jutkiewicz made the breakthrough with a forceful header.
But Danny Hylton extended his run as the competition's top scorer with a dramatic goal on the counter-attack to tee up a replay at the Recreation Ground, only for Jutkiewicz to break the spirited visitors' resolve with a goal at the death.
Meanwhile, Michael Appleton earned his first win as Blackburn boss as Rovers reached the last 16 of the FA Cup for the first time since 2009 with a 3-0 win over Derby County.
Colin Kazim-Richards opened the scoring at Pride Park a minute before half-time and Scott Dann added a second goal as the midway-point of the second period approached. Jordan Rhodes settled the tie shortly afterwards with his 22nd goal of the season.
It was Blackburn's first victory in four games since beating Bristol City in the third round and Appleton's maiden success since taking charge three matches ago.
Chris Dagnall struck his third goal in as many games to send Barnsley through to the fifth round following a 1-0 win over npower Championship rivals Hull.
Hull dominated possession and carried the more potent threat, but Barnsley improved after the break and Dagnall, who scored twice in the recent derby win over Leeds, converted the winner from a rebound at the far post in the 70th minute.
Elsewhere, Chris Wood earned Leicester an FA Cup replay as his 82nd-minute goal left managerless Huddersfield with a 1-1 draw.
The Terriers dominated for large periods and led through Lee Novak's penalty but Wood, who bagged a brace in a 6-1 win over Town on New Year's Day, ensured the sides will meet again at the King Power Stadium.
- Sports & Recreation
- Michael Appleton