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The Dutch midfielder's 79th minute header was a cruel blow for Gladbach, who had looked on track to end their recent run of poor form with a hard-fought win.
"It was important to bring a bit of emotion onto the pitch today, and we did that and got the fans behind us," Gladbach midfielder Christoph Kramer.
"A point is no use to us really, but the way we played was a good sign after our bloodless performances in recent weeks."
Gladbach dropped out of the top four last weekend, after which it was announced that coach Dieter Hecking would leave the club at the end of the season.
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They have won just one of their last seven games, and are three points adrift of fourth-placed Eintracht Frankfurt in the chase for a Champions League spot.
Florian Neuhaus' opener just after half-time breathed life into what had been a dreary game. A mistake by Bremen captain Max Kruse allowed Gladbach to counter-attack, and the 22-year-old rounded the goalkeeper and finished from a tight angle.
Yet with 11 minutes to go, Yuya Osako picked out Klaassen, who sent a powerful header past Gladbach goalkeeper Yann Sommer.
The draw crowns a successful week for Bremen, who won their German Cup quarter-final in midweek to set up a semi-final tie with Bayern Munich later this month.
They are eighth in the Bundesliga, one point behind Hoffenheim, who thrashed Augsburg 4-0 in Sunday's early game.
A second-half hat-trick from Ishak Belfodil left Hoffenheim just one point behind the Europa League places, hot on the heels of Wolfsburg and Borussia Moenchengladbach.
Coach Julian Nagelsmann, 31, has repeatedly spoken of his desire to guide the club back into European competition before he leaves for RB Leipzig this summer, and Belfodil's star performance moved Hoffenheim a step closer on Sunday.
"We still need to win all our games because the teams above us have been unbelievably consistent this year," Nagelsmann told Sky.
Andrej Kramaric scored his 15th goal of the season to give the visitors an early lead, sweeping the ball in on the rebound after Belfodil had forced a save from Augsburg goalkeeper Gregor Kobel.
Kobel later denied Kramaric with a stunning reflex save, but he could do nothing to stop Belfodil's diving header just after the hour mark.
Belfodil added another on 74 minutes, breaking away on the counter-attack and finding the bottom corner with a smart finish.
The Algerian completed his hat-trick with an almost identical goal eight minutes from time.