The U’s got off to a dream start when the lively Jevani Brown stole possession in midfield and burst forward before feeding Maris, who scored with an angled first-time shot into the far corner.
Cambridge doubled their advantage in the 27th minute when David Amoo seized on an error from Zoumana Bakayogo to run clear on goal and beat Ben Garratt with a powerful low shot.
But Crewe were back in it three minutes later, when Callum Ainley’s through ball found Chris Dagnall, who rounded David Forde to score.
Crewe pressed for an equaliser after the break, with Leon Legge making a last-ditch tackle to deny Dagnall.
But it was the home side who wrapped up the points in the 70th minute when Amoo delivered a cross which was flicked in by Uche Ikpeazu.
Cambridge substitute Jabo Ibehre was sent off in stoppage time following an off-the-ball incident.
Nantes are reportedly plotting a summer swoop to sign defender Douglas Pereira dos Santos from Barcelona.