Then, on 20 minutes, Rangers full-back Steven Whittaker whipped in a dangerous cross that foxed Artur Boruc before coming off the upright and the visitors scrambled the ball away to safety.
Celtic's Scott McDonald had a penalty appeal waved away when he went down in the box with Whittaker in close attendance.
Just as the match seemed to be settling, with Celtic enjoying more possession and slowing play down in midfield, Rangers took the lead on 37 minutes.
Miller was the creator, taking the ball away from goal to the bye-line before producing a perfect cut-back across the six-yard box, where Davis beat Shaun Maloney at the back post to slam in his first Old Firm goal past Boruc.
As the visitors pressed to end the half level, scorer Davis turned stopper when he hacked Vennegoor of Hesselink's downward header off the line with Alexander beaten.
Bougherra (left) excelled in defence as Rangers shut out Celtic
Rangers found themselves being pressed deeper into their own half after the restart, with Maloney, ineffectual in his first league start since December, beginning to swing in more crosses on top of Alexander and Gary Caldwell advancing to provide extra support in midfield.
But, for all their possession, Celtic were lucky not to find themselves 2-0 down just before the hour mark.
All it took was a simple turn and through-pass from Miller to Boyd, who skipped past the challenge of O'Dea.
But, with only Boruc to beat, the prolific striker hit his shot straight at the Pole's chest and, minutes later, Boyd fired wide when set up by Davis.
The tempo lifted and the game opened up going into the last 15 minutes and Aiden McGeady - on for Maloney - jinked past a defender to crack a shot off the outside of the post.
But, with David Weir and Bougherra resolute for Rangers in the centre-back positions, Celtic could ill-afford to pass up any glimpse of goal and the Ibrox side recorded what could prove to be a crucial victory in terms of the destination of Scottish Premier League championship.
Rangers: Alexander, Dailly, Bougherra, Weir, Whittaker, Davis, Mendes, Edu, Steven Smith (Lafferty 69), Boyd (McCulloch 85), Miller. Subs Not Used: McGregor, Novo, Velicka, Fleck, Wilson.
Booked: Steven Smith, Miller, Lafferty.
Goals: Davis 37.
Celtic: Boruc, Hinkel, Loovens, Caldwell, O'Dea (Naylor 81), Nakamura, Hartley, Crosas, Maloney (McGeady 62), Vennegoor of Hesselink (Samaras 76), McDonald. Subs Not Used: Mark Brown, Misun, Flood, Robson.
Booked: Hartley, Caldwell, Naylor.
Ref: C Thomson
BBC Sport Player Rater man of the match: Rangers' Steven Davis 8.27 (on 90 minutes).
Please note that you can still give the players marks out of 10 on BBC Sport's Player Rater after the match has finished.
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